├── pdf ├── 05C20-Cut.pdf ├── 05C05-Branch.pdf ├── 05C05-Tree.pdf ├── 05C05-Trie.pdf ├── 05C20-Flow.pdf ├── 05C38-Girth.pdf ├── 05C99-Block.pdf ├── 05C99-Bridge.pdf ├── 05C99-Loop.pdf ├── 05D99-Tight.pdf ├── 05B35-Matroid.pdf ├── 05C05-AVLTree.pdf ├── 05C05-NullTree.pdf ├── 05C40-Valency.pdf ├── 05C70-Matching.pdf ├── 05C99-Subgraph.pdf ├── 05D10-Coloring.pdf ├── 05D15-Saturate.pdf ├── 05A10-Factorial.pdf ├── 05A15-CrazyDice.pdf ├── 05A15-Derangement.pdf ├── 05A18-BellNumber.pdf ├── 05A19-PascalsRule.pdf ├── 05B15-LatinSquare.pdf ├── 05B15-MagicSquare.pdf ├── 05B35-Polymatroid.pdf ├── 05C05-Antichain.pdf ├── 05C05-BinaryTree.pdf ├── 05C05-KonigsLemma.pdf ├── 05C05-SuslinTree.pdf ├── 05C12-Diameter12.pdf ├── 05C15-KempeChain.pdf ├── 05C15-PropertyB.pdf ├── 05C20-Tournament.pdf ├── 05C25-CayleyGraph.pdf ├── 05C38-LaverTable.pdf ├── 05C38-SimplePath.pdf ├── 05C40-Cutvertex.pdf ├── 05C45-Traceable.pdf ├── 05C65-Hypergraph.pdf ├── 05C75-Multigraph.pdf ├── 05C75-Pseudograph.pdf ├── 05C99-NullGraph.pdf ├── 05C99-Subdivision.pdf ├── 05A05-RuleOfProduct.pdf ├── 05A18-BellsTriangle.pdf ├── 05B10-DifferenceSet.pdf ├── 05B15-MagicConstant.pdf ├── 05C05-BalancedTree.pdf ├── 05C05-SpanningTree.pdf ├── 05C10-CrossingLemma.pdf ├── 05C12-HammingMetric.pdf ├── 05C15-KpartiteGraph.pdf ├── 05C15-TaitColoring.pdf ├── 05C20-DirectedGraph.pdf ├── 05C38-AcyclicGraph.pdf ├── 05C45-OresTheorem1.pdf ├── 05C70-EdgeCovering.pdf ├── 05C70-TutteTheorem.pdf ├── 05C78-LabeledGraph.pdf ├── 05C99-InfiniteGraph.pdf ├── 05C99-MinorofAGraph.pdf ├── 05C99-OrderofAGraph.pdf ├── 05C99-SizeofAGraph.pdf ├── 05C99-TuransTheorem.pdf ├── 05D10-RamseyNumbers.pdf ├── 05A10-BirthdayProblem.pdf ├── 05A10-DoubleFactorial.pdf ├── 05A10-FallingFactorial.pdf ├── 05A10-HosoyasTriangle.pdf ├── 05A10-PascalsRuleProof.pdf ├── 05A16-TakeuchiFunction.pdf ├── 05A16-TakeuchiNumber.pdf ├── 05A17-PrimePartition.pdf ├── 05A18-ArrowsRelation.pdf ├── 05B35-GeometricLattice.pdf ├── 05B35-ProjectiveBasis.pdf ├── 05C05-TreeTraversals.pdf ├── 05C10-CrossingNumber.pdf ├── 05C12-DistanceinAGraph.pdf ├── 05C20-DeBruijnDigraph.pdf ├── 05C20-LabelledDigraph.pdf ├── 05C38-VeblensTheorem.pdf ├── 05C40-ConnectedGraph.pdf ├── 05C40-kconnectedGraph.pdf ├── 05C45-FleurysAlgorithm.pdf ├── 05C45-HamiltonianCycle.pdf ├── 05C45-HamiltonianGraph.pdf ├── 05C45-HamiltonianPath.pdf ├── 05C45-Hypohamiltonian.pdf ├── 05C50-AlonChungLemma.pdf ├── 05C69-MantelsTheorem.pdf ├── 05C70-PetersenTheorem.pdf ├── 05C99-Edgecontraction.pdf ├── 05C99-HandshakeLemma.pdf ├── 05C99-Homeomorphism1.pdf ├── 05C99-PoincareFormula.pdf ├── 05C99-WagnersTheorem.pdf ├── 05D10-RamseysTheorem1.pdf ├── 05-00-MultiindexNotation.pdf ├── 05A05-PermutationPattern.pdf ├── 05A10-MultinomialTheorem.pdf ├── 05A15-StirlingPolynomial.pdf ├── 05A17-PartitionFunction.pdf ├── 05A18-ExpectationExample.pdf ├── 05A19-ProofOfPascalsRule.pdf ├── 05B15-GraecoLatinSquares.pdf ├── 05B35-IncidenceGeometry.pdf ├── 05C05-CompleteBinaryTree.pdf ├── 05C05-DigitalSearchTree.pdf ├── 05C10-KuratowskisTheorem.pdf ├── 05C40-TuttesWheelTheorem.pdf ├── 05C99-GraphHomeomorphism.pdf ├── 05C99-HarmonicFunction1.pdf ├── 05D10-ErdHosRadoTheorem.pdf ├── 05E30-AssociationScheme.pdf ├── 05A05-PermutationNotation.pdf ├── 05A10-AlternatingFactorial.pdf ├── 05A10-ExponentialFactorial.pdf ├── 05A10-GeneralizedFactorial.pdf ├── 05A10-SingmastersConjecture.pdf ├── 05A19-KonigEgervaryTheorem.pdf ├── 05A19-VandermondeIdentity.pdf ├── 05A30-GaussianPolynomials.pdf ├── 05B15-ExampleOfLatinSquares.pdf ├── 05B15-FranklinMagicSquare.pdf ├── 05B25-TacticalDecomposition.pdf ├── 05C05-MinimumSpanningTree.pdf ├── 05C10-ProofOfCrossingLemma.pdf ├── 05C15-ChromaticPolynomial.pdf ├── 05C38-LosanitschsTriangle.pdf ├── 05C38-ProofOfVeblensTheorem.pdf ├── 05C50-ProofOfAlonChungLemma.pdf ├── 05C75-ProofOfMantelsTheorem.pdf ├── 05C99-NeighborhoodofAVertex.pdf ├── 05C99-ProofOfTuransTheorem.pdf ├── 05C99-ProofOfWagnersTheorem.pdf ├── 05C99-RealizationOfAGraph.pdf ├── 05D10-ProofOfRamseysTheorem.pdf ├── 05D15-HallsMarriageTheorem.pdf ├── 05A10-LeibnizHarmonicTriangle.pdf ├── 05A10-MultinomialTheoremproof.pdf ├── 05C15-ChromaticNumberAndGirth.pdf ├── 05C15-ColoringsOfPlaneGraphs.pdf ├── 05C38-MultiplicativeEncoding.pdf ├── 05C38-closedWalkTrekTrailPath.pdf ├── 05C45-BondyAndChvatalTheorem.pdf ├── 05C50-LaplacianMatrixOfAGraph.pdf ├── 05C99-EulersPolyhedronTheorem.pdf ├── 05-00-EnumerativeCombinatorics.pdf ├── 05A10-CentralBinomialCoefficient.pdf ├── 05A10-PascalsRulebitStringProof.pdf ├── 05B35-MatroidIndependenceAxioms.pdf ├── 05C05-ExampleOfTreesetTheoretic.pdf ├── 05C40-ExamplesOfkconnectedGraphs.pdf ├── 05-00-ExampleOfPigeonholePrinciple.pdf ├── 05-00-MultiindexDerivativeOfAPower.pdf ├── 05A10-LeviCivitaPermutationSymbol.pdf ├── 05A15-WedderburnEtheringtonNumber.pdf ├── 05B35-AxiomaticProjectiveGeometry.pdf ├── 05C20-MaximumFlowminimumCutTheorem.pdf ├── 05C30-NumberOfUnrootedLabeledTrees.pdf ├── 05C80-ExampleOfAProbabilisticProof.pdf ├── 05C99-UniformlyLocallyFiniteGraph.pdf ├── 05D15-HallsMarriageTheoremProofOf.pdf ├── 05-00-CountingCompositionsOfAnInteger.pdf ├── 05-00-TopicEntryOnDiscreteMathematics.pdf ├── 05A05-OnelineNotationForPermutations.pdf ├── 05A10-InductiveProofOfBinomialTheorem.pdf ├── 05A15-StirlingNumbersOfTheFirstKind.pdf ├── 05A15-StirlingNumbersOfTheSecondKind.pdf ├── 05A99-PrincipleOfInclusionexclusion.pdf ├── 05C15-ProofOfVizingsTheoremforGraphs.pdf ├── 05C38-ProofOfChromaticNumberAndGirth.pdf ├── 05C45-ProofOfBondyAndChvatalTheorem.pdf ├── 05C50-AlgebraicConnectivityOfAGraph.pdf ├── 05C75-AlternateProofOfMantelsTheorem.pdf ├── 05D15-SystemOfDistinctRepresentatives.pdf ├── 05E05-ElementarySymmetricPolynomial.pdf ├── 05A10-CombinationsWithRepeatedElements.pdf ├── 05A10-SumOfPowersOfBinomialCoefficients.pdf ├── 05C05-ProofThatomegaHasTheTreeProperty.pdf ├── 05C70-MaximalBipartiteMatchingAlgorithm.pdf ├── 05A10-SomeFormulasInvolvingRisingFactorial.pdf ├── 05A99-PrincipleOfInclusionexclusionProofOf.pdf ├── 05B15-ConstructionOfMagicSquareOfOddLength.pdf ├── 05C40-ThomassensTheoremOn3connectedGraphs.pdf ├── 05C69-IndependentSetAndIndependenceNumber.pdf ├── 05C75-AlternativeDefinitionOfAMultigraph.pdf ├── 05A10-GeneratingFunctionForTheCatalanNumbers.pdf ├── 05C75-MooreGraphsOfD2AreVvalentAndOrderIsV21.pdf ├── 05A05-ProofOfRecurrencesForDerangementNumbers.pdf ├── 05A10-DivisibilityOfCentralBinomialCoefficient.pdf ├── 05A10-UpperAndLowerBoundsToBinomialCoefficient.pdf ├── 05B20-ProofThatHadamardMatrixHasOrder1Or2Or4n.pdf ├── 05C05-WeightbalancedBinaryTreesAreUltrametric.pdf ├── 05C50-IncidenceMatrixWithRespectToAnOrientation.pdf ├── 05C70-MaximalMatchingminimalEdgeCoveringTheorem.pdf ├── 05E05-ReductionAlgorithmForSymmetricPolynomials.pdf ├── 05A10-DivisibilityOfPrimepowerBinomialCoefficients.pdf ├── 05A15-EnumerationOfLatticeWalksderivationOfFormulas.pdf ├── 05B25-CriterionForANearlinearSpaceBeingALinearSpace.pdf ├── 05C69-SizeOfMaximalIndependentSetAndChromaticNumber.pdf ├── 05D10-RamseytheoreticProofOfTheErdHosSzekeresTheorem.pdf ├── 05A10-ProofOfUpperAndLowerBoundsToBinomialCoefficient.pdf ├── 05A19-GeneratingFunctionForTheReciprocalCatalanNumbers.pdf ├── 05E05-ElementarySymmetricPolynomialInTermsOfPowerSums.pdf ├── 05A19-PascalsTriangleIsSymmetricalAlongItsCentralColumn.pdf ├── 05E05-AlgebraicIndependenceOfElementarySymmetricPolynomials.pdf ├── 05C83-EquivalenceBetweenTheMinorAndTopologicalMinorOfK5OrK331.pdf ├── 05A19-GeneratingFunctionForTheReciprocalCentralBinomialCoefficients.pdf ├── 05E05-ProofOfAlgebraicIndependenceOfElementarySymmetricPolynomials.pdf ├── 05A15-DerivationOfTheGeneratingSeriesForTheStirlingNumbersOfTheSecondKind.pdf ├── 05A10-DerivationOfGeneratingFunctionForTheReciprocalCentralBinomialCoefficients.pdf └── nocando ├── README.md ├── 05C10-CrossingNumber.tex ├── 05C05-NullTree.tex ├── 05C05-DigitalSearchTree.tex ├── 05C40-Cutvertex.tex ├── 05C10-CrossingLemma.tex ├── 05C15-PropertyB.tex ├── 05B35-Polymatroid.tex ├── 05C45-OresTheorem1.tex ├── 05C05-BalancedTree.tex ├── 05C75-Pseudograph.tex ├── 05C45-HamiltonianPath.tex ├── 05C45-Traceable.tex ├── 05B15-LatinSquare.tex ├── 05C38-ProofOfVeblensTheorem.tex ├── 05B15-MagicSquare.tex ├── 05C45-PetersenGraph.tex ├── 05C45-HamiltonianCycle.tex ├── 05C45-BondyAndChvatalTheorem.tex ├── 05C99-NeighborhoodofAVertex.tex ├── 05C99-Loop.tex ├── 05B15-GraecoLatinSquares.tex ├── 05A19-PascalsRule.tex ├── 05C05-LeafNodeofATree.tex ├── 05C05-CompleteBinaryTree.tex ├── 05C05-InternalNodeofATree.tex ├── 05C99-TuransTheorem.tex ├── 05A19-ProofOfPascalsRule.tex ├── 05B15-ExampleOfLatinSquares.tex ├── 05C15-CompleteBipartiteGraph.tex ├── 05C69-Clique.tex ├── 05C99-Bridge.tex ├── 05A10-PascalsRuleProof.tex ├── 05C40-TuttesWheelTheorem.tex ├── 05-00-ExampleOfPigeonholePrinciple.tex ├── 05C99-InfiniteGraph.tex ├── 05C05-ParentNodeinATree.tex ├── 05C05-SuslinTree.tex ├── 05C38-AcyclicGraph.tex ├── 05E05-ElementarySymmetricPolynomial.tex ├── 05C05-ChildNodeofATree.tex ├── 05C05-AVLTree.tex ├── 05C05-SpanningTree.tex ├── 05C69-MantelsTheorem.tex ├── 05A10-Factorial.tex ├── 05C38-VeblensTheorem.tex ├── 05C45-EulerCircuit.tex ├── 05C99-NullGraph.tex ├── 05C45-Hypohamiltonian.tex ├── 05D15-SystemOfDistinctRepresentatives.tex ├── 05C10-KuratowskisTheorem.tex ├── 05C05-MinimumSpanningTree.tex ├── 05C99-RealizationOfAGraph.tex ├── 05C99-WagnersTheorem.tex ├── 05E05-AlgebraicIndependenceOfElementarySymmetricPolynomials.tex ├── 05-00-TopicEntryOnDiscreteMathematics.tex ├── 05C15-ChromaticNumber.tex ├── 05C99-SizeofAGraph.tex ├── 05C99-MinorofAGraph.tex ├── 05A30-GaussianPolynomials.tex ├── 05C05-ExtendedBinaryTree.tex ├── 05C70-EdgeCovering.tex ├── 05C99-Homeomorphism1.tex ├── 05C05-ExternalPathLength.tex ├── 05C25-HyperbolicGroup.tex ├── 05C99-EulersPolyhedronTheorem.tex ├── 05A10-UpperAndLowerBoundsToBinomialCoefficient.tex ├── 05C83-EquivalenceBetweenTheMinorAndTopologicalMinorOfK5OrK331.tex ├── 05A10-DoubleFactorial.tex ├── 05C38-EulerPath.tex ├── 05C50-AlonChungLemma.tex ├── 05D10-Coloring.tex ├── 05C99-Edgecontraction.tex ├── 05C05-Antichain.tex ├── 05C69-SizeOfMaximalIndependentSetAndChromaticNumber.tex ├── 05A10-SomeFormulasInvolvingRisingFactorial.tex ├── 05C99-Block.tex ├── 05A10-ExponentialFactorial.tex ├── 05C99-OrderofAGraph.tex ├── 05C99-CompleteGraph.tex ├── 05C38-SimplePath.tex ├── 05C45-FleurysAlgorithm.tex ├── 05A10-InductiveProofOfBinomialTheorem.tex ├── 05C69-IndependentSetAndIndependenceNumber.tex ├── 05C70-Matching.tex ├── 05C45-ProofOfBondyAndChvatalTheorem.tex ├── 05C38-Girth.tex ├── 05C90-HasseDiagram.tex ├── 05C75-Multigraph.tex ├── 05C05-MinimumWeightedPathLength.tex ├── 05C99-UniformlyLocallyFiniteGraph.tex ├── 05C99-WheelGraph.tex ├── 05B15-MagicConstant.tex ├── 05C99-PoincareFormula.tex ├── 05C15-KpartiteGraph.tex ├── 05C99-KneserGraphs.tex ├── 05C70-PetersenTheorem.tex ├── 05D10-ErdHosRadoTheorem.tex ├── 05C50-LaplacianMatrixOfAGraph.tex ├── 05C05-RootofATree.tex ├── 05C05-WeightedPathLength.tex └── 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https://raw.githubusercontent.com/planetmath/05_Combinatorics/HEAD/pdf/05A10-DerivationOfGeneratingFunctionForTheReciprocalCentralBinomialCoefficients.pdf -------------------------------------------------------------------------------- /README.md: -------------------------------------------------------------------------------- 1 | # fem2016 2 | 3 | [![Join the chat at https://gitter.im/planetmath/05_Combinatorics](https://badges.gitter.im/planetmath/05_Combinatorics.svg)](https://gitter.im/planetmath/05_Combinatorics?utm_source=badge&utm_medium=badge&utm_campaign=pr-badge&utm_content=badge) 4 | 2016 Edition of the Free Encyclopedia of Mathematics (top-level repo) 5 | -------------------------------------------------------------------------------- /05C10-CrossingNumber.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{CrossingNumber} 4 | \pmcreated{2013-03-22 13:20:34} 5 | \pmmodified{2013-03-22 13:20:34} 6 | \pmowner{bbukh}{348} 7 | \pmmodifier{bbukh}{348} 8 | \pmtitle{crossing number} 9 | \pmrecord{5}{33858} 10 | \pmprivacy{1} 11 | \pmauthor{bbukh}{348} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C10} 15 | \pmrelated{PlanarGraph} 16 | 17 | \endmetadata 18 | 19 | \usepackage{amssymb} 20 | \usepackage{amsmath} 21 | \usepackage{amsfonts} 22 | 23 | \DeclareMathOperator{\crn}{cr} 24 | \begin{document} 25 | The \emph{crossing number} $\crn(G)$ of a \PMlinkname{graph}{Graph} $G$ is the minimal number of crossings among all embeddings of $G$ in the plane. 26 | %%%%% 27 | %%%%% 28 | \end{document} 29 | -------------------------------------------------------------------------------- /05C05-NullTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{NullTree} 4 | \pmcreated{2013-03-22 12:31:16} 5 | \pmmodified{2013-03-22 12:31:16} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{null tree} 9 | \pmrecord{6}{32760} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Data Structure} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmsynonym{null subtree}{NullTree} 16 | \pmsynonym{empty tree}{NullTree} 17 | \pmsynonym{empty subtree}{NullTree} 18 | \pmrelated{Tree} 19 | 20 | \endmetadata 21 | 22 | \usepackage{amssymb} 23 | \usepackage{amsmath} 24 | \usepackage{amsfonts} 25 | \begin{document} 26 | \PMlinkescapeword{nodes} 27 | 28 | A \emph{null tree} is simply a tree with zero nodes. 29 | %%%%% 30 | %%%%% 31 | \end{document} 32 | -------------------------------------------------------------------------------- /05C05-DigitalSearchTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{DigitalSearchTree} 4 | \pmcreated{2013-03-22 12:28:31} 5 | \pmmodified{2013-03-22 12:28:31} 6 | \pmowner{Logan}{6} 7 | \pmmodifier{Logan}{6} 8 | \pmtitle{digital search tree} 9 | \pmrecord{8}{32682} 10 | \pmprivacy{1} 11 | \pmauthor{Logan}{6} 12 | \pmtype{Data Structure} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmclassification{msc}{68P10} 16 | \pmclassification{msc}{68P20} 17 | \pmrelated{DigitalTree} 18 | 19 | \endmetadata 20 | 21 | \usepackage{amssymb} 22 | \usepackage{amsmath} 23 | \usepackage{amsfonts} 24 | \begin{document} 25 | A \emph{digital search tree} is a tree which stores strings internally so that there is no need for extra leaf nodes to store the strings. 26 | %%%%% 27 | %%%%% 28 | \end{document} 29 | -------------------------------------------------------------------------------- /05C40-Cutvertex.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Cutvertex} 4 | \pmcreated{2013-03-22 12:31:38} 5 | \pmmodified{2013-03-22 12:31:38} 6 | \pmowner{yark}{2760} 7 | \pmmodifier{yark}{2760} 8 | \pmtitle{cut-vertex} 9 | \pmrecord{6}{32767} 10 | \pmprivacy{1} 11 | \pmauthor{yark}{2760} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C40} 15 | \pmsynonym{cutvertex}{Cutvertex} 16 | \pmsynonym{cut vertex}{Cutvertex} 17 | %\pmkeywords{component} 18 | \pmrelated{Bridge} 19 | \pmrelated{ConnectedGraph} 20 | \pmrelated{Block} 21 | 22 | \endmetadata 23 | 24 | 25 | \begin{document} 26 | A \emph{cut-vertex} of a graph 27 | is a vertex whose deletion increases the number of components of the graph. 28 | The edge analogue of a cut-vertex is a bridge. 29 | %%%%% 30 | %%%%% 31 | \end{document} 32 | -------------------------------------------------------------------------------- /05C10-CrossingLemma.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{CrossingLemma} 4 | \pmcreated{2013-03-22 13:20:37} 5 | \pmmodified{2013-03-22 13:20:37} 6 | \pmowner{bbukh}{348} 7 | \pmmodifier{bbukh}{348} 8 | \pmtitle{crossing lemma} 9 | \pmrecord{6}{33859} 10 | \pmprivacy{1} 11 | \pmauthor{bbukh}{348} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C10} 15 | \pmrelated{PlanarGraph} 16 | 17 | \endmetadata 18 | 19 | \usepackage{amssymb} 20 | \usepackage{amsmath} 21 | \usepackage{amsfonts} 22 | 23 | \DeclareMathOperator{\crn}{cr} 24 | \begin{document} 25 | The crossing number of a \PMlinkname{graph}{Graph} $G$ with $n$ \PMlinkname{vertices}{Graph} and $m\geq 4n$ \PMlinkname{edges}{Graph} is 26 | \begin{equation*} 27 | \crn(G)\geq \frac{1}{64}\frac{m^3}{n^2}. 28 | \end{equation*} 29 | %%%%% 30 | %%%%% 31 | \end{document} 32 | -------------------------------------------------------------------------------- /05C15-PropertyB.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{PropertyB} 4 | \pmcreated{2013-03-22 13:39:10} 5 | \pmmodified{2013-03-22 13:39:10} 6 | \pmowner{bbukh}{348} 7 | \pmmodifier{bbukh}{348} 8 | \pmtitle{property B} 9 | \pmrecord{6}{34306} 10 | \pmprivacy{1} 11 | \pmauthor{bbukh}{348} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C15} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amssymb} 19 | \usepackage{amsmath} 20 | \usepackage{amsfonts} 21 | 22 | \makeatletter 23 | \@ifundefined{bibname}{}{\renewcommand{\bibname}{References}} 24 | \makeatother 25 | \begin{document} 26 | A hypergraph $G$ is said to possess \emph{property B} if it $2$-colorable, i.e., its vertices can be colored in two colors, so that no edge of $G$ is monochromatic. 27 | 28 | The property was named after Felix Bernstein by E. W. Miller. 29 | %%%%% 30 | %%%%% 31 | \end{document} 32 | -------------------------------------------------------------------------------- /05B35-Polymatroid.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Polymatroid} 4 | \pmcreated{2013-03-22 13:56:43} 5 | \pmmodified{2013-03-22 13:56:43} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{polymatroid} 9 | \pmrecord{4}{34707} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05B35} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amssymb} 19 | \usepackage{amsmath} 20 | \usepackage{amsfonts} 21 | \begin{document} 22 | The \emph{polymatroid} defined by a given matroid $(E,r)$ is the set of 23 | all functions $w:E\to\mathbb{R}$ such that 24 | $$w(e)\ge 0\qquad\text{for all }e\in E$$ 25 | $$\sum_{e\in S}w(e)\le r(S)\qquad\text{for all }S\subset E\;.$$ 26 | Polymatroids are related to the convex polytopes seen in linear programming, 27 | and have similar uses. 28 | %%%%% 29 | %%%%% 30 | \end{document} 31 | -------------------------------------------------------------------------------- /05C45-OresTheorem1.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{OresTheorem} 4 | \pmcreated{2013-03-22 11:52:40} 5 | \pmmodified{2013-03-22 11:52:40} 6 | \pmowner{Koro}{127} 7 | \pmmodifier{Koro}{127} 8 | \pmtitle{Ore's theorem} 9 | \pmrecord{10}{30473} 10 | \pmprivacy{1} 11 | \pmauthor{Koro}{127} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmclassification{msc}{18D20} 16 | \pmrelated{HamiltonianGraph} 17 | \pmrelated{BondyAndChvatalTheorem} 18 | 19 | \endmetadata 20 | 21 | \usepackage{amssymb} 22 | \usepackage{amsmath} 23 | \usepackage{amsfonts} 24 | \usepackage{graphicx} 25 | %%%%\usepackage{xypic} 26 | \begin{document} 27 | Let $G$ be a simple graph of order $n\ge 3$ such that, for every pair of distinct non adjacent vertices $u$ and $v$, $\deg(u)+\deg(v)\ge n$. 28 | Then $G$ is a Hamiltonian graph. 29 | %%%%% 30 | %%%%% 31 | %%%%% 32 | %%%%% 33 | \end{document} 34 | -------------------------------------------------------------------------------- /05C05-BalancedTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{BalancedTree} 4 | \pmcreated{2013-03-22 12:29:11} 5 | \pmmodified{2013-03-22 12:29:11} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{balanced tree} 9 | \pmrecord{7}{32706} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Data Structure} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmrelated{Tree} 16 | \pmrelated{BalancedBinaryTree} 17 | \pmrelated{Heap} 18 | \pmrelated{BinaryTree} 19 | 20 | \endmetadata 21 | 22 | \usepackage{amssymb} 23 | \usepackage{amsmath} 24 | \usepackage{amsfonts} 25 | \begin{document} 26 | A \emph{balanced tree } is a rooted tree where no leaf is much farther 27 | away from the root than any other leaf. 28 | Different balancing \PMlinkescapetext{schemes} allow different definitions of "much farther" and 29 | different amounts of work to keep them balanced. For an example, see binary tree. 30 | 31 | 32 | 33 | %%%%% 34 | %%%%% 35 | \end{document} 36 | -------------------------------------------------------------------------------- /05C75-Pseudograph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Pseudograph} 4 | \pmcreated{2013-03-22 11:58:00} 5 | \pmmodified{2013-03-22 11:58:00} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{pseudograph} 9 | \pmrecord{9}{30781} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C75} 15 | %\pmkeywords{graph} 16 | \pmrelated{Graph} 17 | \pmrelated{Multigraph} 18 | \pmrelated{LoopOfAGraph} 19 | \pmrelated{Subgraph} 20 | \pmrelated{GraphHomomorphism} 21 | 22 | \endmetadata 23 | 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | \usepackage{graphicx} 28 | %%%\usepackage{xypic} 29 | \begin{document} 30 | A \emph{pseudograph} is a graph that allows both parallel edges and loops. 31 | Formally, $G=(V, E)$ is a pseudograph , if $E$ is a multiset $(V^{(2)}, f)$ 32 | where $V^{(2)}$ is the set of unordered pairs of $V$. 33 | %%%%% 34 | %%%%% 35 | %%%%% 36 | \end{document} 37 | -------------------------------------------------------------------------------- /05C45-HamiltonianPath.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{HamiltonianPath} 4 | \pmcreated{2013-03-22 11:52:46} 5 | \pmmodified{2013-03-22 11:52:46} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{Hamiltonian path} 9 | \pmrecord{10}{30475} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmclassification{msc}{46L05} 16 | \pmclassification{msc}{82-00} 17 | \pmclassification{msc}{83-00} 18 | \pmclassification{msc}{81-00} 19 | \pmrelated{HamiltonianCycle} 20 | \pmrelated{HamiltonianGraph} 21 | \pmrelated{PetersensGraph} 22 | \pmrelated{Traceable} 23 | 24 | \endmetadata 25 | 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | \usepackage{graphicx} 30 | %%%%\usepackage{xypic} 31 | \begin{document} 32 | Let $G$ be a graph. A path on $G$ that includes every vertex exactly once is called a \emph{Hamiltonian path}. 33 | %%%%% 34 | %%%%% 35 | %%%%% 36 | %%%%% 37 | \end{document} 38 | -------------------------------------------------------------------------------- /05C45-Traceable.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Traceable} 4 | \pmcreated{2013-03-22 11:52:52} 5 | \pmmodified{2013-03-22 11:52:52} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{traceable} 9 | \pmrecord{9}{30477} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmclassification{msc}{03B22} 16 | \pmclassification{msc}{03-00} 17 | \pmclassification{msc}{03-01} 18 | \pmrelated{HamiltonianPath} 19 | \pmrelated{HamiltonianCycle} 20 | \pmrelated{HamiltonianGraph} 21 | \pmrelated{PetersensGraph} 22 | 23 | \endmetadata 24 | 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | \usepackage{graphicx} 29 | %%%%\usepackage{xypic} 30 | \begin{document} 31 | Let $G$ be a graph. If $G$ has a Hamiltonian path, we say that $G$ is \emph{traceable}. 32 | 33 | Not every traceable graph is Hamiltonian. As an example consider Petersen's graph. 34 | %%%%% 35 | %%%%% 36 | %%%%% 37 | %%%%% 38 | \end{document} 39 | -------------------------------------------------------------------------------- /05B15-LatinSquare.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{LatinSquare} 4 | \pmcreated{2013-03-22 12:14:34} 5 | \pmmodified{2013-03-22 12:14:34} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{Latin square} 9 | \pmrecord{7}{31624} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05B15} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amsmath} 19 | \begin{document} 20 | A \emph{Latin square} of \PMlinkescapetext{order}$n$ is an $n\times n$ array such that each column and each row are made with the same $n$ symbols, using every one exactly once . 21 | 22 | Examples. 23 | \begin{equation*} 24 | \left(\begin{array}{cccc} 25 | a & b & c & d\\ 26 | c & d & a &b\\ 27 | d & c & b & a\\ 28 | b & a & d & c 29 | \end{array}\right) 30 | \qquad 31 | \left(\begin{array}{cccc} 32 | 1 & 2& 3& 4\\ 33 | 4 & 3 & 2 & 1\\ 34 | 2 & 1 & 4 & 3\\ 35 | 3 & 4 & 1 & 2 36 | \end{array}\right) 37 | \end{equation*} 38 | %%%%% 39 | %%%%% 40 | %%%%% 41 | \end{document} 42 | -------------------------------------------------------------------------------- /05C38-ProofOfVeblensTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ProofOfVeblensTheorem} 4 | \pmcreated{2013-03-22 13:56:51} 5 | \pmmodified{2013-03-22 13:56:51} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{proof of Veblen's theorem} 9 | \pmrecord{4}{34711} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Proof} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C38} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amssymb} 19 | \usepackage{amsmath} 20 | \usepackage{amsfonts} 21 | \begin{document} 22 | The proof is very easy by induction on the number of elements of 23 | the set $E$ of edges. 24 | If $E$ is empty, then all the vertices have degree zero, which is even. 25 | Suppose $E$ is nonempty. 26 | If the graph contains no cycle, then some vertex has degree $1$, which is odd. 27 | Finally, if the graph does contain a cycle $C$, then every vertex has 28 | the same degree mod $2$ with respect to $E-C$, as it has with respect 29 | to $E$, and we can conclude by induction. 30 | %%%%% 31 | %%%%% 32 | \end{document} 33 | -------------------------------------------------------------------------------- /05B15-MagicSquare.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{MagicSquare} 4 | \pmcreated{2013-03-22 12:14:39} 5 | \pmmodified{2013-03-22 12:14:39} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{magic square} 9 | \pmrecord{6}{31626} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05B15} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amsmath} 19 | \begin{document} 20 | A magic square of order $n$ is an $n\times n$ array using each one of the numbers $1,2,3,\ldots,n^2$ once and such that the sum of the numbers in each row, column or main diagonal is the same. 21 | 22 | Example: 23 | \begin{equation*} 24 | \begin{pmatrix} 25 | 8 & 1 & 6\\ 26 | 3 & 5 & 7\\ 27 | 4 & 9 & 2 28 | \end{pmatrix} 29 | \end{equation*} 30 | 31 | It's easy to prove that the sum is always $\frac{1}{2}n(n^2+1)$. So in the example with $n=3$ the sum is always $\frac{1}{2}(3\times 10)=15$. 32 | 33 | One way to generalize this concept is to allow any numbers in the entries, instead of $1,2,\ldots,n$. 34 | %%%%% 35 | %%%%% 36 | %%%%% 37 | \end{document} 38 | -------------------------------------------------------------------------------- /05C45-PetersenGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{PetersenGraph} 4 | \pmcreated{2013-03-22 11:52:55} 5 | \pmmodified{2013-03-22 11:52:55} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{Petersen graph} 9 | \pmrecord{10}{30478} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmclassification{msc}{46L05} 16 | \pmclassification{msc}{82-00} 17 | \pmclassification{msc}{83-00} 18 | \pmclassification{msc}{81-00} 19 | \pmrelated{Traceable} 20 | \pmrelated{HamiltonianPath} 21 | \pmrelated{HamiltonianGraph} 22 | 23 | \endmetadata 24 | 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | \usepackage{graphicx} 29 | %%%%\usepackage{xypic} 30 | \begin{document} 31 | \emph{Petersen's graph}. An example of graph that is traceable but not Hamiltonian. That is, it has a Hamiltonian path but doesn't have a Hamiltonian cycle. 32 | 33 | \begin{center} 34 | \includegraphics{petersen} 35 | \end{center} 36 | 37 | This is also the canonical example of a hypohamiltonian graph. 38 | %%%%% 39 | %%%%% 40 | %%%%% 41 | %%%%% 42 | \end{document} 43 | -------------------------------------------------------------------------------- /05C45-HamiltonianCycle.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{HamiltonianCycle} 4 | \pmcreated{2013-03-22 11:52:49} 5 | \pmmodified{2013-03-22 11:52:49} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{Hamiltonian cycle} 9 | \pmrecord{11}{30476} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmclassification{msc}{81T70} 16 | \pmclassification{msc}{81T05} 17 | \pmclassification{msc}{81T25} 18 | \pmclassification{msc}{81T20} 19 | \pmclassification{msc}{81T75} 20 | %\pmkeywords{Graph} 21 | \pmrelated{HamiltonianGraph} 22 | \pmrelated{HamiltonianPath} 23 | \pmrelated{GraphTheory} 24 | \pmrelated{Traceable} 25 | 26 | \endmetadata 27 | 28 | \usepackage{amssymb} 29 | \usepackage{amsmath} 30 | \usepackage{amsfonts} 31 | \usepackage{graphicx} 32 | %%%%\usepackage{xypic} 33 | \begin{document} 34 | Let $G$ be a graph. If there is a cycle visiting all vertices of $G$ exactly once, we say that the cycle is a \emph{Hamiltonian cycle}. A graph having a Hamiltonian cycle is called a \emph{Hamiltonian graph}. 35 | %%%%% 36 | %%%%% 37 | %%%%% 38 | %%%%% 39 | \end{document} 40 | -------------------------------------------------------------------------------- /05C45-BondyAndChvatalTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{BondyAndChvatalTheorem} 4 | \pmcreated{2013-03-22 11:52:57} 5 | \pmmodified{2013-03-22 11:52:57} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{Bondy and Chv\'atal theorem} 9 | \pmrecord{9}{30479} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmclassification{msc}{81P99} 16 | \pmclassification{msc}{81S30} 17 | \pmclassification{msc}{81S99} 18 | \pmclassification{msc}{81-00} 19 | \pmclassification{msc}{81S05} 20 | \pmclassification{msc}{81P15} 21 | \pmrelated{HamiltonianGraph} 22 | \pmrelated{OresTheorem} 23 | 24 | \endmetadata 25 | 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | %\usepackage{graphicx} 30 | %%%%%\usepackage{xypic} 31 | \begin{document} 32 | \textbf{Bondy and Chv\'atal's theorem.}\\ 33 | Let $G$ be a graph of order $n\ge 3$ and suppose that $u$ and $v$ are distinct non adjacent vertices such that $\deg(u)+\deg(v)\ge n$. 34 | 35 | Then $G$ is Hamiltonian if and only if $G+uv$ is Hamiltonian. 36 | %%%%% 37 | %%%%% 38 | %%%%% 39 | %%%%% 40 | \end{document} 41 | -------------------------------------------------------------------------------- /05C99-NeighborhoodofAVertex.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{NeighborhoodofAVertex} 4 | \pmcreated{2013-03-22 11:58:03} 5 | \pmmodified{2013-03-22 11:58:03} 6 | \pmowner{digitalis}{76} 7 | \pmmodifier{digitalis}{76} 8 | \pmtitle{neighborhood (of a vertex)} 9 | \pmrecord{10}{30785} 10 | \pmprivacy{1} 11 | \pmauthor{digitalis}{76} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmsynonym{neighborhood}{NeighborhoodofAVertex} 16 | %\pmkeywords{vertex} 17 | %\pmkeywords{graph} 18 | \pmrelated{Graph} 19 | 20 | \endmetadata 21 | 22 | \usepackage{amssymb} 23 | \usepackage{amsmath} 24 | \usepackage{amsfonts} 25 | \usepackage{graphicx} 26 | %%%\usepackage{xypic} 27 | \begin{document} 28 | For a graph $G$, the set of vertices adjacent to a vertex $x \in G$, the \emph{neighborhood} of $x$, is denoted by $\Gamma(x)$. Occasionally one calls $\Gamma(x)$ the \emph{open} neighborhood of $x$, and $\Gamma \cup \{x\}$ the \emph{closed} neighborhood of $x$. 29 | 30 | 31 | \footnotesize{Adapted with permission of the author from \emph{Modern Graph Theory} by B\'{e}la Bollob\'{a}s, published by Springer-Verlag New York, Inc., 1998.} 32 | %%%%% 33 | %%%%% 34 | %%%%% 35 | \end{document} 36 | -------------------------------------------------------------------------------- /05C99-Loop.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Loop} 4 | \pmcreated{2013-03-22 12:14:08} 5 | \pmmodified{2013-03-22 12:14:08} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{loop} 9 | \pmrecord{10}{31615} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmclassification{msc}{20N05} 16 | \pmrelated{Graph} 17 | \pmrelated{Pseudograph} 18 | \pmrelated{Quasigroup} 19 | 20 | \endmetadata 21 | 22 | \usepackage{amssymb} 23 | \usepackage{amsmath} 24 | \usepackage{amsfonts} 25 | \begin{document} 26 | \PMlinkescapeword{joins} 27 | \PMlinkescapeword{contain} 28 | In graph theory, a \emph{loop} is an edge which joins a vertex 29 | to itself, rather than to some other vertex. By definition, 30 | a graph cannot contain a loop; a pseudograph, however, may contain 31 | both multiple edges and multiple loops. Note that by some definitions, 32 | a multigraph may contain multiple edges and no loops, while other texts 33 | define a multigraph as a graph 34 | allowing multiple edges and multiple loops. 35 | 36 | In algebra, a \emph{loop} is a quasigroup which contains an identity element. 37 | %%%%% 38 | %%%%% 39 | %%%%% 40 | \end{document} 41 | -------------------------------------------------------------------------------- /05B15-GraecoLatinSquares.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{GraecoLatinSquares} 4 | \pmcreated{2013-03-22 12:14:36} 5 | \pmmodified{2013-03-22 12:14:36} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{Graeco-Latin squares} 9 | \pmrecord{9}{31625} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05B15} 15 | \pmdefines{join} 16 | 17 | \endmetadata 18 | 19 | \usepackage{amsmath} 20 | \begin{document} 21 | Let $A=(a_{ij})$ and $B=(b_{ij})$ be two $n\times n$ matrices. We define their \emph{\PMlinkescapetext{join}} as the matrix whose $(i,j)$th entry is the pair $(a_{ij},b_{ij})$. 22 | 23 | A \emph{Graeco-Latin square} is then the join of two Latin squares. 24 | 25 | The name comes from Euler's use of Greek and Latin letters to differentiate the entries on each array. 26 | 27 | An example of Graeco-Latin square: 28 | \begin{equation*} 29 | \begin{pmatrix} 30 | a\alpha & b\beta & c\gamma & d\delta\\ 31 | d\gamma & c\delta & b\alpha & a\beta\\ 32 | b\delta & a\gamma & d\beta & c\alpha\\ 33 | c\beta & d\alpha & a\delta & b\gamma 34 | \end{pmatrix} 35 | \end{equation*} 36 | %%%%% 37 | %%%%% 38 | %%%%% 39 | \end{document} 40 | -------------------------------------------------------------------------------- /05A19-PascalsRule.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{PascalsRule} 4 | \pmcreated{2013-03-22 11:46:44} 5 | \pmmodified{2013-03-22 11:46:44} 6 | \pmowner{KimJ}{5} 7 | \pmmodifier{KimJ}{5} 8 | \pmtitle{Pascal's rule} 9 | \pmrecord{10}{30246} 10 | \pmprivacy{1} 11 | \pmauthor{KimJ}{5} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A19} 15 | %\pmkeywords{number theory combinatorics} 16 | \pmrelated{BinomialCoefficient} 17 | \pmrelated{VandermondeIdentity} 18 | \pmrelated{PascalsTriangle} 19 | \pmrelated{TheoremOfThePrimalRay} 20 | \pmrelated{Mm2} 21 | \pmrelated{Mm} 22 | \pmrelated{LeTheoremeDuRayonPrimal} 23 | \pmrelated{LeDeuxiemeTheoremeDuRayonPrimal} 24 | \pmrelated{AProofOfGoldbachConjecture} 25 | \pmrelated{AProofOfDePolignacConjectures} 26 | \pmrelated{FermatGhanouchiSeriesAmazingFermatGhanouchiSequ} 27 | 28 | \endmetadata 29 | 30 | \usepackage{amssymb} 31 | \usepackage{amsmath} 32 | \usepackage{amsfonts} 33 | \usepackage{graphicx} 34 | %%%%\usepackage{xypic} 35 | \begin{document} 36 | Pascal's rule is the binomial identity 37 | \[ \binom{n}{k} + \binom{n}{k-1} = \binom{n+1}{k} \] 38 | where $1 \leq k \leq n$ and $\binom{n}{k}$ is the binomial coefficient. 39 | %%%%% 40 | %%%%% 41 | %%%%% 42 | %%%%% 43 | \end{document} 44 | -------------------------------------------------------------------------------- /05C05-LeafNodeofATree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{LeafNodeofATree} 4 | \pmcreated{2013-03-22 12:30:28} 5 | \pmmodified{2013-03-22 12:30:28} 6 | \pmowner{akrowne}{2} 7 | \pmmodifier{akrowne}{2} 8 | \pmtitle{leaf node (of a tree)} 9 | \pmrecord{5}{32741} 10 | \pmprivacy{1} 11 | \pmauthor{akrowne}{2} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmsynonym{leaf node}{LeafNodeofATree} 16 | \pmsynonym{leaf}{LeafNodeofATree} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | 24 | %\usepackage{psfrag} 25 | %\usepackage{graphicx} 26 | %%\usepackage{xypic} 27 | \xyoption{all} 28 | \usepackage{color} 29 | \begin{document} 30 | A \emph{leaf} of a tree is any node which has degree of exactly 1. Put another way, a leaf node of a rooted tree is any node which has no child nodes. 31 | 32 | \begin{center} 33 | 34 | $$\xymatrix{ 35 | & \bullet \ar@{-}[dl] \ar@{-}[dr] & & & \\ 36 | {\color{red}\bullet} & & \bullet \ar@{-}[dr]\ar@{-}[dl] & & \\ 37 | & \bullet \ar@{-}[dl] & & {\color{red}\bullet} & \\ 38 | {\color{red}\bullet} & & & & }$$ 39 | 40 | {\tiny Figure: A tree with leaf nodes highlighted in red.} 41 | \end{center} 42 | %%%%% 43 | %%%%% 44 | \end{document} 45 | -------------------------------------------------------------------------------- /05C05-CompleteBinaryTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{CompleteBinaryTree} 4 | \pmcreated{2013-03-22 12:30:14} 5 | \pmmodified{2013-03-22 12:30:14} 6 | \pmowner{akrowne}{2} 7 | \pmmodifier{akrowne}{2} 8 | \pmtitle{complete binary tree} 9 | \pmrecord{7}{32736} 10 | \pmprivacy{1} 11 | \pmauthor{akrowne}{2} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmsynonym{complete}{CompleteBinaryTree} 16 | \pmrelated{ExtendedBinaryTree} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | 24 | %\usepackage{psfrag} 25 | %\usepackage{graphicx} 26 | %%%\usepackage{xypic} 27 | \begin{document} 28 | A \emph{complete binary tree} is a binary tree with the additional property that every node must have exactly two ``children'' if an internal node, and zero children if a leaf node. 29 | 30 | More precisely: for our base case, the complete binary tree of exactly one node is simply the tree consisting of that node by itself. The property of being ``complete'' is preserved if, at each step, we expand the tree by connecting exactly zero or two individual nodes (or complete binary trees) to any node in the tree (but both must be connected to the same node.) 31 | %%%%% 32 | %%%%% 33 | \end{document} 34 | -------------------------------------------------------------------------------- /05C05-InternalNodeofATree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{InternalNodeofATree} 4 | \pmcreated{2013-03-22 12:30:25} 5 | \pmmodified{2013-03-22 12:30:25} 6 | \pmowner{akrowne}{2} 7 | \pmmodifier{akrowne}{2} 8 | \pmtitle{internal node (of a tree)} 9 | \pmrecord{6}{32740} 10 | \pmprivacy{1} 11 | \pmauthor{akrowne}{2} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmsynonym{internal node}{InternalNodeofATree} 16 | 17 | \endmetadata 18 | 19 | \usepackage{amssymb} 20 | \usepackage{amsmath} 21 | \usepackage{amsfonts} 22 | 23 | %\usepackage{psfrag} 24 | %\usepackage{graphicx} 25 | \usepackage{color} 26 | %%\usepackage{xypic} 27 | \xyoption{all} 28 | \begin{document} 29 | An \emph{internal node} of a tree is any node which has degree greater than one. Or, phrased in rooted tree terminology, the internal nodes of a tree are the nodes which have at least one child node. 30 | 31 | \begin{center} 32 | 33 | $$\xymatrix{ 34 | & {\color{red}\bullet} \ar@{-}[dl] \ar@{-}[dr] & & & \\ 35 | \bullet & & {\color{red}\bullet} \ar@{-}[dr]\ar@{-}[dl] & & \\ 36 | & {\color{red}\bullet} \ar@{-}[dl] & & \bullet & \\ 37 | \bullet & & & & }$$ 38 | 39 | {\tiny Figure: A tree with internal nodes highlighted in red.} 40 | \end{center} 41 | %%%%% 42 | %%%%% 43 | \end{document} 44 | -------------------------------------------------------------------------------- /05C99-TuransTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TuransTheorem} 4 | \pmcreated{2013-03-22 12:44:10} 5 | \pmmodified{2013-03-22 12:44:10} 6 | \pmowner{mathwizard}{128} 7 | \pmmodifier{mathwizard}{128} 8 | \pmtitle{Turan's theorem} 9 | \pmrecord{5}{33037} 10 | \pmprivacy{1} 11 | \pmauthor{mathwizard}{128} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | A graph with $n$ vertices, which contains no $p$-\PMlinkid{clique}{1757} with $p\geq 2$, has at most 41 | $$\left(1-\frac{1}{p-1}\right)\frac{n^2}{2}$$ 42 | edges. 43 | %%%%% 44 | %%%%% 45 | \end{document} 46 | -------------------------------------------------------------------------------- /05A19-ProofOfPascalsRule.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ProofOfPascalsRule} 4 | \pmcreated{2013-03-22 15:03:11} 5 | \pmmodified{2013-03-22 15:03:11} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{proof of Pascal's rule} 9 | \pmrecord{5}{36770} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Proof} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A19} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amsmath} 19 | \begin{document} 20 | The definition of $\binom{n}{k}$ is the number of $k$-subsets out from an $n$-set. Using this combinatorial meaning the proof is straightforward. 21 | 22 | Let $x$ a distinct element from the $n$-set. As previously stated, $\binom{n}{k}$ counts the number of subsets with $k$ elements, chosen from the set with $n$ elements. Now, some of these subsets will contain $x$ and some others don't. 23 | 24 | The number of $k$-subsets not containing $x$ is $\binom{n-1}{k}$, since we need to choose $k$ elements from the $n-1$ elements different from $x$. 25 | 26 | The number of $k$-subsets containing $x$ is $\binom{n-1}{k-1}$, because if it is given that $x$ is in the subset, we only need to choose the remaining $k-1$ elements from the $n-1$ elements that are different from $x$. 27 | 28 | Thus 29 | \[ 30 | \binom{n}{k} = \binom{n-1}{k} + \binom{n-1}{k-1}. 31 | \] 32 | %%%%% 33 | %%%%% 34 | \end{document} 35 | -------------------------------------------------------------------------------- /05B15-ExampleOfLatinSquares.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ExampleOfLatinSquares} 4 | \pmcreated{2013-03-22 13:06:41} 5 | \pmmodified{2013-03-22 13:06:41} 6 | \pmowner{jgade}{861} 7 | \pmmodifier{jgade}{861} 8 | \pmtitle{example of Latin squares} 9 | \pmrecord{6}{33539} 10 | \pmprivacy{1} 11 | \pmauthor{jgade}{861} 12 | \pmtype{Example} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05B15} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | 33 | 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | \begin{document} 37 | It is easily shown that the multiplication table (Cayley-table) of a group has exactly these properties and thus are Latin squares. However the converse is not true, ie. not all Latin squares are multiplication tables for a group (the smallest counter example is a Latin square of order 5). 38 | %%%%% 39 | %%%%% 40 | \end{document} 41 | -------------------------------------------------------------------------------- /05C15-CompleteBipartiteGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{CompleteBipartiteGraph} 4 | \pmcreated{2013-03-22 12:17:13} 5 | \pmmodified{2013-03-22 12:17:13} 6 | \pmowner{yark}{2760} 7 | \pmmodifier{yark}{2760} 8 | \pmtitle{complete bipartite graph} 9 | \pmrecord{7}{31784} 10 | \pmprivacy{1} 11 | \pmauthor{yark}{2760} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C15} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amssymb} 19 | \usepackage{amsmath} 20 | \usepackage{amsfonts} 21 | 22 | %%\usepackage{xypic} 23 | \begin{document} 24 | The \emph{complete bipartite graph} $K_{n,m}$ is a graph with two sets of vertices, one with $n$ members and one with $m$, such that each vertex in one set is adjacent to every vertex in the other set and to no vertex in its own set. As the name implies, $K_{n,m}$ is bipartite. 25 | 26 | Examples of complete bipartite graphs: 27 | 28 | $K_{2,5}$: 29 | 30 | $$\xymatrix{ 31 | & C \\ 32 | A \ar@{-}[ur] \ar@{-}[r] \ar@{-}[dr] \ar@{-}[ddr] \ar@{-}[dddr] & D \\ 33 | & E \\ 34 | B \ar@{-}[uuur] \ar@{-}[uur] \ar@{-}[ur] \ar@{-}[r] \ar@{-}[dr] & F \\ 35 | & G 36 | }$$ 37 | 38 | $K_{3,3}$: 39 | 40 | $$\xymatrix{ 41 | A \ar@{-}[r] \ar@{-}[dr] \ar@{-}[ddr] & D \\ 42 | B \ar@{-}[ur] \ar@{-}[r] \ar@{-}[dr] & E \\ 43 | C \ar@{-}[uur] \ar@{-}[ur] \ar@{-}[r] & F 44 | }$$ 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05C69-Clique.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Clique} 4 | \pmcreated{2013-03-22 12:30:53} 5 | \pmmodified{2013-03-22 12:30:53} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{clique} 9 | \pmrecord{13}{32752} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C69} 15 | %\pmkeywords{subgraph} 16 | %\pmkeywords{maximal complete subgraph} 17 | \pmrelated{IndependentSetAndIndependenceNumber} 18 | \pmdefines{clique number} 19 | \pmdefines{maximum clique} 20 | 21 | \endmetadata 22 | 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | \begin{document} 27 | \PMlinkescapeword{maximal}A maximal \PMlinkid{complete}{1757} subgraph of a graph is a \emph{clique}, and the \emph{clique number} $\omega(G)$ of a graph $G$ is the \PMlinkescapephrase{maximal order} maximal order of a clique in $G$. Simply, $\omega(G)$ is the maximal order of a \PMlinkescapetext{complete} subgraph of $G$. Some authors however define a clique as any \PMlinkescapetext{complete} subgraph of $G$ and refer to the other definition as \textit{maximum clique}. 28 | 29 | 30 | \footnotesize{Adapted with permission of the author from \emph{\PMlinkescapetext{Modern Graph Theory}} by B\'{e}la Bollob\'{a}s, published by Springer-Verlag New York, Inc., 1998.} 31 | %%%%% 32 | %%%%% 33 | \end{document} 34 | -------------------------------------------------------------------------------- /05C99-Bridge.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Bridge} 4 | \pmcreated{2013-03-22 12:31:41} 5 | \pmmodified{2013-03-22 12:31:41} 6 | \pmowner{digitalis}{76} 7 | \pmmodifier{digitalis}{76} 8 | \pmtitle{bridge} 9 | \pmrecord{5}{32768} 10 | \pmprivacy{1} 11 | \pmauthor{digitalis}{76} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmrelated{Cutvertex} 16 | \pmrelated{ConnectedGraph} 17 | \pmrelated{Block} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | \begin{document} 43 | A \emph{bridge} of a graph $G$ is an edge whose deletion increases the number of components of $G$. The vertex analogue of a bridge is a cutvertex. 44 | %%%%% 45 | %%%%% 46 | \end{document} 47 | -------------------------------------------------------------------------------- /05A10-PascalsRuleProof.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{PascalsRuleProof} 4 | \pmcreated{2013-03-22 11:47:14} 5 | \pmmodified{2013-03-22 11:47:14} 6 | \pmowner{akrowne}{2} 7 | \pmmodifier{akrowne}{2} 8 | \pmtitle{Pascal's rule proof} 9 | \pmrecord{10}{30259} 10 | \pmprivacy{1} 11 | \pmauthor{akrowne}{2} 12 | \pmtype{Proof} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A10} 15 | \pmclassification{msc}{81T13} 16 | \pmclassification{msc}{53C80} 17 | \pmclassification{msc}{82-00} 18 | \pmclassification{msc}{83-00} 19 | \pmclassification{msc}{81-00} 20 | 21 | \endmetadata 22 | 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | \usepackage{graphicx} 27 | %%%%\usepackage{xypic} 28 | \begin{document} 29 | We need to show 30 | \begin{eqnarray*} 31 | \binom{n}{k} + \binom{n}{k-1} & = & \binom{n+1}{k} 32 | \end{eqnarray*} 33 | Let us begin by writing the left-hand side as $$ \frac{n!}{k!(n-k)!} + \frac{n!}{(k-1)!(n-(k-1))!}$$ 34 | Getting a common denominator and simplifying, we have 35 | \begin{eqnarray*} 36 | \frac{n!}{k!(n-k)!} + \frac{n!}{(k-1)!(n-k+1)!} & = & \frac{(n-k+1)n!}{(n-k+1)k!(n-k)!}+\frac{kn!}{k(k-1)!(n-k+1)!} \\ 37 | & = & \frac{(n-k+1)n!+kn!}{k!(n-k+1)!} \\ 38 | & = & \frac{(n+1)n!}{k!((n+1)-k)!} \\ 39 | & = & \frac{(n+1)!}{k!((n+1)-k)!} \\ 40 | & = & \binom{n+1}{k} 41 | \end{eqnarray*} 42 | %%%%% 43 | %%%%% 44 | %%%%% 45 | %%%%% 46 | \end{document} 47 | -------------------------------------------------------------------------------- /05C40-TuttesWheelTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TuttesWheelTheorem} 4 | \pmcreated{2013-03-22 13:11:06} 5 | \pmmodified{2013-03-22 13:11:06} 6 | \pmowner{lieven}{1075} 7 | \pmmodifier{lieven}{1075} 8 | \pmtitle{Tutte's wheel theorem} 9 | \pmrecord{4}{33633} 10 | \pmprivacy{1} 11 | \pmauthor{lieven}{1075} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C40} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | Every \PMlinkname{$3$-connected}{KConnectedGraph} simple graph can be constructed starting from a wheel graph by repeatedly either adding an edge between two non-adjacent vertices or splitting a vertex. 41 | %%%%% 42 | %%%%% 43 | \end{document} 44 | -------------------------------------------------------------------------------- /05-00-ExampleOfPigeonholePrinciple.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ExampleOfPigeonholePrinciple} 4 | \pmcreated{2013-03-22 12:41:32} 5 | \pmmodified{2013-03-22 12:41:32} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{example of pigeonhole principle} 9 | \pmrecord{8}{32972} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Example} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05-00} 15 | 16 | \endmetadata 17 | 18 | \usepackage{graphicx} 19 | %%%\usepackage{xypic} 20 | \usepackage{bbm} 21 | \usepackage{amsthm} 22 | \newtheorem*{thm}{Theorem} 23 | \newcommand{\Z}{\mathbbmss{Z}} 24 | \newcommand{\C}{\mathbbmss{C}} 25 | \newcommand{\R}{\mathbbmss{R}} 26 | \newcommand{\Q}{\mathbbmss{Q}}{ 27 | \newcommand{\mathbb}[1]{\mathbbmss{#1}} 28 | \newcommand{\figura}[1]{\begin{center}\includegraphics{#1}\end{center}} 29 | 30 | \begin{document} 31 | A \PMlinkescapetext{simple} example. 32 | \begin{thm} For any set of $8$ integers, there exist at least two of them 33 | whose difference is divisible by $7$. 34 | \end{thm} 35 | 36 | \small 37 | \begin{proof} 38 | The residue classes modulo $7$ are $0,1,2,3,4,5,6$. 39 | We have seven \PMlinkescapetext{classes} and eight integers. So it must be the case that 2 integers fall on the same 40 | residue class, and therefore their difference will be divisible by $7$. 41 | \end{proof} 42 | %%%%% 43 | %%%%% 44 | \end{document} 45 | -------------------------------------------------------------------------------- /05C99-InfiniteGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{InfiniteGraph} 4 | \pmcreated{2013-03-22 16:00:57} 5 | \pmmodified{2013-03-22 16:00:57} 6 | \pmowner{sjm1979}{13837} 7 | \pmmodifier{sjm1979}{13837} 8 | \pmtitle{infinite graph} 9 | \pmrecord{5}{38051} 10 | \pmprivacy{1} 11 | \pmauthor{sjm1979}{13837} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | An \PMlinkescapetext{{\sl infinite graph\/}} is graph such that the edge and vertex sets each have infinite cardinality. An example of an infinite graph is that of $\mathbb{Z}^2$ as stipulated in the entry locally finite graph. 42 | %%%%% 43 | %%%%% 44 | \end{document} 45 | -------------------------------------------------------------------------------- /05C05-ParentNodeinATree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ParentNodeinATree} 4 | \pmcreated{2013-03-22 12:30:32} 5 | \pmmodified{2013-03-22 12:30:32} 6 | \pmowner{akrowne}{2} 7 | \pmmodifier{akrowne}{2} 8 | \pmtitle{parent node (in a tree)} 9 | \pmrecord{5}{32742} 10 | \pmprivacy{1} 11 | \pmauthor{akrowne}{2} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmsynonym{parent node}{ParentNodeinATree} 16 | \pmsynonym{parent}{ParentNodeinATree} 17 | \pmrelated{ChildNodeOfATree} 18 | 19 | \endmetadata 20 | 21 | \usepackage{amssymb} 22 | \usepackage{amsmath} 23 | \usepackage{amsfonts} 24 | 25 | %\usepackage{psfrag} 26 | %\usepackage{graphicx} 27 | %%\usepackage{xypic} 28 | \xyoption{all} 29 | \usepackage{color} 30 | \begin{document} 31 | A \emph{parent} node $P$ of a node $C$ in a tree is the first node which lies along the path from $C$ to the root of the tree, $R$. 32 | 33 | Drawn in the canonical root-at-top manner, the parent node of a node $C$ in a tree is simply the node immediately above $C$ which is connected to it. 34 | 35 | \begin{center} 36 | 37 | $$\xymatrix{ 38 | & \bullet \ar@{-}[dl] \ar@{-}[dr] & & & \\ 39 | \bullet & & {\color{red}\bullet} \ar@{-}[dr]\ar@{-}[dl] & & \\ 40 | & \bullet \ar@{-}[dl] & & {\color{blue}\bullet} & \\ 41 | \bullet & & & & }$$ 42 | 43 | {\tiny Figure: A node (blue) and its parent (red.)} 44 | \end{center} 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05C05-SuslinTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SuslinTree} 4 | \pmcreated{2013-03-22 12:52:42} 5 | \pmmodified{2013-03-22 12:52:42} 6 | \pmowner{Henry}{455} 7 | \pmmodifier{Henry}{455} 8 | \pmtitle{Suslin tree} 9 | \pmrecord{4}{33220} 10 | \pmprivacy{1} 11 | \pmauthor{Henry}{455} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmclassification{msc}{03E05} 16 | \pmrelated{TreeSetTheoretic} 17 | \pmrelated{AronszajnTree} 18 | \pmrelated{Aronszajn} 19 | \pmdefines{Suslin tree} 20 | 21 | \endmetadata 22 | 23 | % this is the default PlanetMath preamble. as your knowledge 24 | % of TeX increases, you will probably want to edit this, but 25 | % it should be fine as is for beginners. 26 | 27 | % almost certainly you want these 28 | \usepackage{amssymb} 29 | \usepackage{amsmath} 30 | \usepackage{amsfonts} 31 | 32 | % used for TeXing text within eps files 33 | %\usepackage{psfrag} 34 | % need this for including graphics (\includegraphics) 35 | %\usepackage{graphicx} 36 | % for neatly defining theorems and propositions 37 | %\usepackage{amsthm} 38 | % making logically defined graphics 39 | %%%\usepackage{xypic} 40 | 41 | % there are many more packages, add them here as you need them 42 | 43 | % define commands here 44 | %\PMlinkescapeword{theory} 45 | \begin{document} 46 | An Aronszajn tree is a Suslin tree iff it has no uncountable antichains. 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /05C38-AcyclicGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{AcyclicGraph} 4 | \pmcreated{2013-03-22 12:30:39} 5 | \pmmodified{2013-03-22 12:30:39} 6 | \pmowner{Logan}{6} 7 | \pmmodifier{Logan}{6} 8 | \pmtitle{acyclic graph} 9 | \pmrecord{8}{32746} 10 | \pmprivacy{1} 11 | \pmauthor{Logan}{6} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C38} 15 | \pmsynonym{acyclic}{AcyclicGraph} 16 | \pmsynonym{DAG}{AcyclicGraph} 17 | \pmrelated{Graph} 18 | \pmrelated{Cycle} 19 | \pmrelated{BetheLattice} 20 | \pmdefines{directed acyclic graph} 21 | 22 | \endmetadata 23 | 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | \usepackage[all]{xy} 28 | \begin{document} 29 | Any graph that contains no cycles is an \emph{acyclic graph}. A directed acyclic graph is often called a DAG for short. 30 | 31 | For example, the following graph and digraph are acyclic. 32 | 33 | $$ 34 | \begin{array}{cc} 35 | \xymatrix{&A\ar@{-}[dl]\ar@{-}[dr]\\B&&C} 36 | \quad 37 | & 38 | \quad 39 | \xymatrix{&A\ar[dr]\\B\ar[ur]\ar[rr]&&C} 40 | \end{array} 41 | $$ 42 | 43 | In contrast, the following graph and digraph are \emph{not} acyclic, because 44 | each contains a cycle. 45 | 46 | $$ 47 | \begin{array}{cc} 48 | \xymatrix{&A\ar@{-}[dl]\ar@{-}[dr]\\B\ar@{-}[rr]&&C} 49 | \quad 50 | & 51 | \quad 52 | \xymatrix{&A\ar[dr]\\B\ar[ur]&&C\ar[ll]} 53 | \end{array} 54 | $$ 55 | %%%%% 56 | %%%%% 57 | \end{document} 58 | -------------------------------------------------------------------------------- /05E05-ElementarySymmetricPolynomial.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ElementarySymmetricPolynomial} 4 | \pmcreated{2013-03-22 12:09:01} 5 | \pmmodified{2013-03-22 12:09:01} 6 | \pmowner{djao}{24} 7 | \pmmodifier{djao}{24} 8 | \pmtitle{elementary symmetric polynomial} 9 | \pmrecord{9}{31340} 10 | \pmprivacy{1} 11 | \pmauthor{djao}{24} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05E05} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amssymb} 19 | \usepackage{amsmath} 20 | \usepackage{amsfonts} 21 | \usepackage{graphicx} 22 | %%%\usepackage{xypic} 23 | \begin{document} 24 | The coefficient of $x^{n-k}$ in the polynomial $(x+t_1) (x+t_2) \cdots (x+t_n)$ is called the $k^\mathrm{th}$ \emph{elementary symmetric polynomial} in the $n$ variables $t_1, \dots, t_n$. The elementary symmetric polynomials can also be constructed by taking the sum of all possible degree $k$ monomials in $t_1,\dots, t_n$ having distinct factors. 25 | 26 | The first few examples are: 27 | \begin{description} 28 | \item[$n=1$:] 29 | $ 30 | \begin{array}{l} 31 | t_1 32 | \end{array} 33 | $ 34 | \item[$n=2$:] 35 | 36 | $ 37 | \begin{array}{l} 38 | t_1 + t_2\\ 39 | t_1 t_2 40 | \end{array} 41 | $ 42 | \item[$n=3$:] 43 | 44 | $ 45 | \begin{array}{l} 46 | t_1 + t_2 + t_3\\ 47 | t_1 t_2 + t_2 t_3 + t_1 t_3\\ 48 | t_1 t_2 t_3 49 | \end{array} 50 | $ 51 | \end{description} 52 | %%%%% 53 | %%%%% 54 | %%%%% 55 | \end{document} 56 | -------------------------------------------------------------------------------- /05C05-ChildNodeofATree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ChildNodeofATree} 4 | \pmcreated{2013-03-22 12:30:37} 5 | \pmmodified{2013-03-22 12:30:37} 6 | \pmowner{akrowne}{2} 7 | \pmmodifier{akrowne}{2} 8 | \pmtitle{child node (of a tree)} 9 | \pmrecord{4}{32744} 10 | \pmprivacy{1} 11 | \pmauthor{akrowne}{2} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmsynonym{child node}{ChildNodeofATree} 16 | \pmsynonym{child}{ChildNodeofATree} 17 | \pmrelated{ParentNodeInATree} 18 | 19 | \endmetadata 20 | 21 | \usepackage{amssymb} 22 | \usepackage{amsmath} 23 | \usepackage{amsfonts} 24 | 25 | %\usepackage{psfrag} 26 | %\usepackage{graphicx} 27 | %%\usepackage{xypic} 28 | \xyoption{all} 29 | \usepackage{color} 30 | \begin{document} 31 | A \emph{child node} $C$ of a node $P$ in a tree is any node connected to $P$ which has a path distance from the root node $R$ which is one greater than the path distance between $P$ and $R$. 32 | 33 | Drawn in the canonical root-at-top manner, a child node of a node $P$ in a tree is simply any node immediately below $P$ which is connected to it. 34 | 35 | \begin{center} 36 | 37 | $$\xymatrix{ 38 | & \bullet \ar@{-}[dl] \ar@{-}[dr] & & & \\ 39 | \bullet & & {\color{blue}\bullet} \ar@{-}[dr]\ar@{-}[dl] & & \\ 40 | & {\color{red}\bullet} \ar@{-}[dl] & & {\color{red}\bullet} & \\ 41 | \bullet & & & & }$$ 42 | 43 | {\tiny Figure: A node (blue) and its children (red.)} 44 | \end{center} 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05C05-AVLTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{AVLTree} 4 | \pmcreated{2013-03-22 13:24:39} 5 | \pmmodified{2013-03-22 13:24:39} 6 | \pmowner{PrimeFan}{13766} 7 | \pmmodifier{PrimeFan}{13766} 8 | \pmtitle{AVL tree} 9 | \pmrecord{9}{33956} 10 | \pmprivacy{1} 11 | \pmauthor{PrimeFan}{13766} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | An \emph{AVL tree} is a balanced binary search tree where the height of the two 41 | subtrees (children) of a node differs by at most one. Look-up, insertion, and 42 | deletion are $O( \ln{n})$, where $n$ is the number of nodes in the tree. 43 | 44 | The structure is named for the inventors, Adelson-Velskii and Landis (1962). 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05C05-SpanningTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SpanningTree} 4 | \pmcreated{2013-03-22 12:29:19} 5 | \pmmodified{2013-03-22 12:29:19} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{spanning tree} 9 | \pmrecord{6}{32709} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmrelated{Tree} 16 | \pmrelated{Graph} 17 | \pmrelated{MinimumSpanningTree} 18 | \pmrelated{DepthFirstSearch2} 19 | \pmrelated{DepthFirstSearch} 20 | 21 | \endmetadata 22 | 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | \usepackage[all]{xy} 27 | \begin{document} 28 | A \emph{spanning tree} of a (connected) graph $G$ is a connected, acyclic subgraph of $G$ that contains all of the vertices of $G$. Below is an example of a spanning tree $T$, where the edges in $T$ are drawn as solid lines and the edges in $G$ but not in $T$ are drawn as dotted lines. 29 | 30 | $$ 31 | \xymatrix{ 32 | \bullet \ar@{.}[dddd] \ar@{-}[ddr] \ar@{.}[rrr] &&& \bullet \ar@{.}[ddll] \ar@{-}[dd] \ar@{.}[drr] \\ 33 | &&&&&\bullet \ar@{-}[dll] \ar@{.}[d] \\ 34 | &\bullet\ar@{-}[ddl]\ar@{-}[dr]\ar@{.}[rr] && \bullet\ar@{-}[dl]\ar@{.}[dr]\ar@{-}[rr] &&\bullet\ar@{-}[dl] \\ 35 | &&\bullet\ar@{.}[dll]\ar@{-}[dr]\ar@{.}[rr]&&\bullet\ar@{.}[dl] \\ 36 | \bullet\ar@{.}[rrr]&&&\bullet 37 | } 38 | $$ 39 | 40 | For any tree there is exactly one spanning tree: the tree itself. 41 | %%%%% 42 | %%%%% 43 | \end{document} 44 | -------------------------------------------------------------------------------- /05C69-MantelsTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{MantelsTheorem} 4 | \pmcreated{2013-03-22 12:31:30} 5 | \pmmodified{2013-03-22 12:31:30} 6 | \pmowner{digitalis}{76} 7 | \pmmodifier{digitalis}{76} 8 | \pmtitle{Mantel's theorem} 9 | \pmrecord{5}{32764} 10 | \pmprivacy{1} 11 | \pmauthor{digitalis}{76} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C69} 15 | \pmclassification{msc}{05C75} 16 | %\pmkeywords{graph} 17 | %\pmkeywords{cycle} 18 | %\pmkeywords{triangle} 19 | \pmrelated{Graph} 20 | \pmrelated{Cycle} 21 | \pmrelated{OrderOfAGraph} 22 | \pmrelated{SizeOfAGraph} 23 | 24 | \endmetadata 25 | 26 | % this is the default PlanetMath preamble. as your knowledge 27 | % of TeX increases, you will probably want to edit this, but 28 | % it should be fine as is for beginners. 29 | 30 | % almost certainly you want these 31 | \usepackage{amssymb} 32 | \usepackage{amsmath} 33 | \usepackage{amsfonts} 34 | 35 | % used for TeXing text within eps files 36 | %\usepackage{psfrag} 37 | % need this for including graphics (\includegraphics) 38 | %\usepackage{graphicx} 39 | % for neatly defining theorems and propositions 40 | %\usepackage{amsthm} 41 | % making logically defined graphics 42 | %%%\usepackage{xypic} 43 | 44 | % there are many more packages, add them here as you need them 45 | 46 | % define commands here 47 | \begin{document} 48 | Every graph of order $n$ and size greater than $\lfloor n^2/4 \rfloor$ contains a triangle (cycle of order 3). 49 | %%%%% 50 | %%%%% 51 | \end{document} 52 | -------------------------------------------------------------------------------- /05A10-Factorial.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Factorial} 4 | \pmcreated{2013-03-22 11:53:58} 5 | \pmmodified{2013-03-22 11:53:58} 6 | \pmowner{yark}{2760} 7 | \pmmodifier{yark}{2760} 8 | \pmtitle{factorial} 9 | \pmrecord{22}{30516} 10 | \pmprivacy{1} 11 | \pmauthor{yark}{2760} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A10} 15 | \pmclassification{msc}{11B65} 16 | \pmclassification{msc}{92-01} 17 | \pmclassification{msc}{92B05} 18 | \pmsynonym{factorial function}{Factorial} 19 | \pmrelated{BinomialCoefficient} 20 | \pmrelated{ExponentialFactorial} 21 | 22 | \endmetadata 23 | 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | \begin{document} 28 | \PMlinkescapeword{word} 29 | 30 | For any non-negative integer $n$, the {\em factorial} of $n$, denoted $n!$, can be defined by 31 | $$n!=\prod_{r=1}^n r$$ 32 | where for $n=0$ the empty product is taken to be $1$. 33 | 34 | Alternatively, the factorial can be defined recursively by $0!=1$ and $n!=n(n-1)!$ for $n>0$. 35 | 36 | $n!$ is equal to the number of permutations of $n$ distinct objects. 37 | For example, there are $5!$ ways to arrange the five letters A, B, C, D and E into a word. 38 | 39 | For every non-negative integer $n$ we have 40 | $$\Gamma(n+1) = n!$$ 41 | where $\Gamma$ is Euler's gamma function. 42 | In this way the notion of factorial can be generalized to all \PMlinkname{complex}{Complex} values except the negative integers. 43 | %%%%% 44 | %%%%% 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05C38-VeblensTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{VeblensTheorem} 4 | \pmcreated{2013-03-22 12:31:27} 5 | \pmmodified{2013-03-22 12:31:27} 6 | \pmowner{digitalis}{76} 7 | \pmmodifier{digitalis}{76} 8 | \pmtitle{Veblen's theorem} 9 | \pmrecord{5}{32763} 10 | \pmprivacy{1} 11 | \pmauthor{digitalis}{76} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C38} 15 | %\pmkeywords{cycle} 16 | %\pmkeywords{graph} 17 | %\pmkeywords{partition} 18 | \pmrelated{Graph} 19 | \pmrelated{Cycle} 20 | \pmrelated{Digraph} 21 | \pmrelated{DegreeOfAVertex} 22 | \pmrelated{DirectedGraph} 23 | 24 | \endmetadata 25 | 26 | % this is the default PlanetMath preamble. as your knowledge 27 | % of TeX increases, you will probably want to edit this, but 28 | % it should be fine as is for beginners. 29 | 30 | % almost certainly you want these 31 | \usepackage{amssymb} 32 | \usepackage{amsmath} 33 | \usepackage{amsfonts} 34 | 35 | % used for TeXing text within eps files 36 | %\usepackage{psfrag} 37 | % need this for including graphics (\includegraphics) 38 | %\usepackage{graphicx} 39 | % for neatly defining theorems and propositions 40 | %\usepackage{amsthm} 41 | % making logically defined graphics 42 | %%%\usepackage{xypic} 43 | 44 | % there are many more packages, add them here as you need them 45 | 46 | % define commands here 47 | \begin{document} 48 | The edge set of a graph can be \PMlinkname{partitioned}{Partition} into cycles if and only if every vertex has even degree. 49 | %%%%% 50 | %%%%% 51 | \end{document} 52 | -------------------------------------------------------------------------------- /05C45-EulerCircuit.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{EulerCircuit} 4 | \pmcreated{2013-03-22 12:02:06} 5 | \pmmodified{2013-03-22 12:02:06} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{Euler circuit} 9 | \pmrecord{12}{31044} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmsynonym{Euler cycle}{EulerCircuit} 16 | %\pmkeywords{Euler path} 17 | %\pmkeywords{Euler circuit} 18 | \pmrelated{EulerPath} 19 | \pmrelated{FleurysAlgorithm} 20 | 21 | \endmetadata 22 | 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | \usepackage{graphicx} 27 | %%%\usepackage{xypic} 28 | \begin{document} 29 | An Euler circuit is a connected graph such that starting at a vertex $a$, one can traverse along every edge of the graph once to each of the other vertices and return to vertex $a$. In other words, an Euler circuit is an Euler path that is a circuit. Thus, using the properties of odd and even \PMlinkid{degree}{788} vertices given in the definition of an Euler path, an Euler circuit exists if and only if every vertex of the graph has an even degree. 30 | 31 | \begin{center} 32 | \includegraphics[width=1in,height=1in]{ecircuit} 33 | \end{center} 34 | 35 | This graph is an Euler circuit as all vertices have degree 2. 36 | 37 | \begin{center} 38 | \includegraphics[width=1in,height=1in]{epath} 39 | \end{center} 40 | 41 | This graph is not an Euler circuit. 42 | %%%%% 43 | %%%%% 44 | %%%%% 45 | \end{document} 46 | -------------------------------------------------------------------------------- /05C99-NullGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{NullGraph} 4 | \pmcreated{2013-03-22 12:48:42} 5 | \pmmodified{2013-03-22 12:48:42} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{null graph} 9 | \pmrecord{11}{33131} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmsynonym{empty graph}{NullGraph} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | \usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | \usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | The null graph is the graph with no vertices or edges. The null graph is the initial object in the category of graphs. 42 | 43 | {\bf Further Reading} 44 | \begin{itemize} 45 | \item Harary, F. and Read, R. ``Is the null-graph a pointless concept?'', Lecture Notes in Mathematics 406 (1974), pp. 37-44. 46 | \end{itemize} 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /05C45-Hypohamiltonian.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Hypohamiltonian} 4 | \pmcreated{2013-03-22 12:25:24} 5 | \pmmodified{2013-03-22 12:25:24} 6 | \pmowner{digitalis}{76} 7 | \pmmodifier{digitalis}{76} 8 | \pmtitle{hypohamiltonian} 9 | \pmrecord{4}{32432} 10 | \pmprivacy{1} 11 | \pmauthor{digitalis}{76} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmclassification{msc}{05C65} 16 | %\pmkeywords{hamiltonian} 17 | \pmrelated{HamiltonianGraph} 18 | \pmrelated{PetersensGraph} 19 | 20 | \endmetadata 21 | 22 | % this is the default PlanetMath preamble. as your knowledge 23 | % of TeX increases, you will probably want to edit this, but 24 | % it should be fine as is for beginners. 25 | 26 | % almost certainly you want these 27 | \usepackage{amssymb} 28 | \usepackage{amsmath} 29 | \usepackage{amsfonts} 30 | 31 | % used for TeXing text within eps files 32 | %\usepackage{psfrag} 33 | % need this for including graphics (\includegraphics) 34 | %\usepackage{graphicx} 35 | % for neatly defining theorems and propositions 36 | %\usepackage{amsthm} 37 | % making logically defined graphics 38 | %%%\usepackage{xypic} 39 | 40 | % there are many more packages, add them here as you need them 41 | 42 | % define commands here 43 | \begin{document} 44 | A graph $G$ is \emph{hypohamiltonian} if $G$ is not Hamiltonian, but $G - v$ is Hamiltonian for each $v \in V$ ($V$ the vertex set of $G$). The smallest hypohamiltonian graph is the Petersen graph, which has ten vertices. 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05D15-SystemOfDistinctRepresentatives.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SystemOfDistinctRepresentatives} 4 | \pmcreated{2013-03-22 12:35:11} 5 | \pmmodified{2013-03-22 12:35:11} 6 | \pmowner{vampyr}{22} 7 | \pmmodifier{vampyr}{22} 8 | \pmtitle{system of distinct representatives} 9 | \pmrecord{4}{32836} 10 | \pmprivacy{1} 11 | \pmauthor{vampyr}{22} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05D15} 15 | \pmsynonym{SDR}{SystemOfDistinctRepresentatives} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | Let $S = \left\{ S_1,S_2,\dots S_n \right\}$ be a finite collection of finite sets. A \emph{system of distinct representatives}, or \emph{SDR}, of $S$ is a set 42 | $$x_1 \in S_1, x_2 \in S_2, \dots x_n \in S_n$$ 43 | such that 44 | $$x_i \neq x_j \text{ whenever } i \neq j$$ 45 | (i.e., each choice must be unique). 46 | %%%%% 47 | %%%%% 48 | \end{document} 49 | -------------------------------------------------------------------------------- /05C10-KuratowskisTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{KuratowskisTheorem} 4 | \pmcreated{2013-03-22 11:57:45} 5 | \pmmodified{2013-03-22 11:57:45} 6 | \pmowner{bbukh}{348} 7 | \pmmodifier{bbukh}{348} 8 | \pmtitle{Kuratowski's theorem} 9 | \pmrecord{12}{30764} 10 | \pmprivacy{1} 11 | \pmauthor{bbukh}{348} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C10} 15 | %\pmkeywords{planar} 16 | \pmrelated{PlanarGraph} 17 | \pmrelated{WagnersTheorem} 18 | 19 | \endmetadata 20 | 21 | \usepackage{amssymb} 22 | \usepackage{amsmath} 23 | \usepackage{amsfonts} 24 | 25 | \makeatletter 26 | \@ifundefined{bibname}{}{\renewcommand{\bibname}{References}} 27 | \makeatother 28 | \begin{document} 29 | A finite graph is planar if and only if it contains no subgraph that is isomorphic to or is a subdivision of $K_5$ or $K_{3,3}$, where $K_5$ is the complete graph of order 5 and $K_{3,3}$ is the complete bipartite graph with 3 vertices in each of the halfs. Wagner's theorem is an equivalent later result. 30 | 31 | \begin{thebibliography}{1} 32 | 33 | \bibitem{cite:kuratowski_planarity} 34 | Kazimierz Kuratowski. 35 | \newblock Sur le probl{\`e}me des courbes gauches en topologie. 36 | \newblock {\em Fund. Math.}, 15:271--283, 1930. 37 | 38 | \end{thebibliography} 39 | 40 | %@ARTICLE{cite:kuratowski_planarity, 41 | % author = {Kazimierz Kuratowski}, 42 | % title = "Sur le Probl{\`e}me des Courbes Gauches en Topologie", 43 | % journal = {Fund. Math.}, 44 | % volume = 15, 45 | % pages = {271--283}, 46 | % year = 1930 47 | %} 48 | %%%%% 49 | %%%%% 50 | %%%%% 51 | \end{document} 52 | -------------------------------------------------------------------------------- /05C05-MinimumSpanningTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{MinimumSpanningTree} 4 | \pmcreated{2013-03-22 12:29:22} 5 | \pmmodified{2013-03-22 12:29:22} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{minimum spanning tree} 9 | \pmrecord{7}{32710} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmsynonym{smallest spanning tree}{MinimumSpanningTree} 16 | \pmrelated{SpanningTree} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | \usepackage[all]{xy} 24 | \begin{document} 25 | Given a graph $G$ with weighted edges, a \emph{minimum spanning tree} is a spanning tree with minimum weight, where the weight of a spanning tree is the sum of the weights of its edges. There may be more than one minimum spanning tree for a graph, since it is the weight of the spanning tree that must be minimum. 26 | 27 | For example, here is a graph $G$ of weighted edges and a minimum spanning tree $T$ for that graph. The edges of $T$ are drawn as solid lines, while edges in $G$ but not in $T$ are drawn as dotted lines. 28 | 29 | $$ 30 | \xymatrix{ 31 | &&\bullet\ar@{-}[dll]|3\ar@{-}[dd]|4\ar@{.}[drr]|7 \\ 32 | \bullet\ar@{.}[dd]|8\ar@{.}[drr]|4&&&&\bullet\ar@{-}[dll]|2\ar@{-}[dd]|5 \\ 33 | &&\bullet\ar@{.}[dll]|5\ar@{-}[dd]|3\ar@{.}[drr]|7 \\ 34 | \bullet\ar@{-}[drr]|2&&&&\bullet\ar@{.}[dll]|6 \\ 35 | &&\bullet 36 | } 37 | $$ 38 | 39 | Prim's algorithm or Kruskal's algorithm can compute the minimum spanning tree of a graph. 40 | %%%%% 41 | %%%%% 42 | \end{document} 43 | -------------------------------------------------------------------------------- /05C99-RealizationOfAGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{RealizationOfAGraph} 4 | \pmcreated{2013-03-22 12:31:58} 5 | \pmmodified{2013-03-22 12:31:58} 6 | \pmowner{rmilson}{146} 7 | \pmmodifier{rmilson}{146} 8 | \pmtitle{realization of a graph} 9 | \pmrecord{7}{32774} 10 | \pmprivacy{1} 11 | \pmauthor{rmilson}{146} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmrelated{Subdivision} 16 | \pmrelated{GraphTopology} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | A \emph{realization of a graph} $G$ in a topological space $X$ is an injective map $\rho:G\to X$ such that $G$ with the graph topology is homeomorphic to $\rho(G)$ with the subspace topology. Of particular interest are realizations of $G$ in 3-dimensional Euclidean space, where each of the edges is a line segment, and where no 4 points of $G$ are coplanar. 43 | %%%%% 44 | %%%%% 45 | \end{document} 46 | -------------------------------------------------------------------------------- /05C99-WagnersTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{WagnersTheorem} 4 | \pmcreated{2013-03-22 12:31:49} 5 | \pmmodified{2013-03-22 12:31:49} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{Wagner's theorem} 9 | \pmrecord{7}{32771} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | %\pmkeywords{planar} 16 | \pmrelated{PlanarGraph} 17 | \pmrelated{KuratowskisTheorem} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | \begin{document} 43 | \newtheorem{thm}{Theorem} 44 | \begin{thm}[Wagner] 45 | A graph is planar if and only if it contains neither $K_5$ nor $K_{3,3}$ as a minor, where $K_5$ is the complete graph of order 5 and $K_{3,3}$ is the complete bipartite graph of order 6. 46 | \end{thm} 47 | Wagner's theorem is \PMlinkescapetext{equivalent} to Kuratowski's theorem. 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /05E05-AlgebraicIndependenceOfElementarySymmetricPolynomials.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{AlgebraicIndependenceOfElementarySymmetricPolynomials} 4 | \pmcreated{2013-03-22 14:49:11} 5 | \pmmodified{2013-03-22 14:49:11} 6 | \pmowner{mclase}{549} 7 | \pmmodifier{mclase}{549} 8 | \pmtitle{algebraic independence of elementary symmetric polynomials} 9 | \pmrecord{5}{36481} 10 | \pmprivacy{1} 11 | \pmauthor{mclase}{549} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05E05} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | \usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \newtheorem*{theorem}{Theorem} 40 | \begin{document} 41 | \begin{theorem} 42 | Let $s_1, s_2, \dots, s_n$ be the elementary symmetric polynomials in $n$ variables $t_1, t_2, \dots, t_n$ over a commutative ring $R$. 43 | Then $s_1, s_2, \dots, s_n$ are algebraically independent elements of 44 | $R[t_1, t_2, \dots, t_n]$ over $R$. 45 | \end{theorem} 46 | %%%%% 47 | %%%%% 48 | \end{document} 49 | -------------------------------------------------------------------------------- /05-00-TopicEntryOnDiscreteMathematics.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{TopicEntryOnDiscreteMathematics} 4 | \pmcreated{2013-03-22 18:00:23} 5 | \pmmodified{2013-03-22 18:00:23} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{topic entry on discrete mathematics} 9 | \pmrecord{4}{40520} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Topic} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05-00} 15 | \pmclassification{msc}{90C99} 16 | \pmclassification{msc}{91A99} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb,amscd} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | \usepackage{mathrsfs} 24 | 25 | % used for TeXing text within eps files 26 | %\usepackage{psfrag} 27 | % need this for including graphics (\includegraphics) 28 | %\usepackage{graphicx} 29 | % for neatly defining theorems and propositions 30 | \usepackage{amsthm} 31 | % making logically defined graphics 32 | %%\usepackage{xypic} 33 | \usepackage{pst-plot} 34 | 35 | % define commands here 36 | \newcommand*{\abs}[1]{\left\lvert #1\right\rvert} 37 | \newtheorem{prop}{Proposition} 38 | \newtheorem{thm}{Theorem} 39 | \newtheorem{ex}{Example} 40 | \newcommand{\real}{\mathbb{R}} 41 | \newcommand{\pdiff}[2]{\frac{\partial #1}{\partial #2}} 42 | \newcommand{\mpdiff}[3]{\frac{\partial^#1 #2}{\partial #3^#1}} 43 | \begin{document} 44 | This is an entry-in-progress for a new topic entry on discrete mathematics. 45 | 46 | \begin{enumerate} 47 | \item Combinatorics 48 | \item Graph theory 49 | \item Cryptography 50 | \item Game theory 51 | \item Mathematical programming 52 | \end{enumerate} 53 | %%%%% 54 | %%%%% 55 | \end{document} 56 | -------------------------------------------------------------------------------- /05C15-ChromaticNumber.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ChromaticNumber} 4 | \pmcreated{2013-05-16 21:23:44} 5 | \pmmodified{2013-05-16 21:23:44} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{unlord}{1} 8 | \pmtitle{chromatic number} 9 | \pmrecord{9}{31764} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{1} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C15} 15 | \pmrelated{ChromaticNumberAndGirth} 16 | \pmrelated{FourColorConjecture} 17 | \pmrelated{ChromaticPolynomial} 18 | \pmrelated{ChromaticNumberOfASpace} 19 | 20 | \endmetadata 21 | 22 | \usepackage{amssymb} 23 | \usepackage{amsmath} 24 | \usepackage{amsfonts} 25 | %%\usepackage{xypic} 26 | \usepackage{color} 27 | \begin{document} 28 | The \emph{chromatic number} of a graph is the minimum number of colours required to colour it. 29 | 30 | Consider the following graph: 31 | 32 | $$\xymatrix{ 33 | {\color{red}A} \ar@{-}[r] & {\color{blue}B} \ar@{-}[r] \ar@{-}[d] & {\color{red}C} \ar@{-}[r] \ar@{-}[dl] & {\color{green}F} \ar@{-}[dl] \\ 34 | & {\color{green}D} \ar@{-}[r] & {\color{blue}E} & }$$ 35 | 36 | This graph has been coloured using $3$ colours. Furthermore, it's clear that it cannot be coloured with fewer than $3$ colours, as well: it contains a subgraph ($BCD$) that is isomorphic to the complete graph of $3$ vertices. As a result, the chromatic number of this graph is indeed $3$. 37 | 38 | This example was easy to solve by inspection. In general, however, finding the chromatic number of a large graph (and, similarly, an optimal colouring) is a very difficult (NP-hard) problem. 39 | %%%%% 40 | %%%%% rerender 41 | \end{document} 42 | -------------------------------------------------------------------------------- /05C99-SizeofAGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SizeofAGraph} 4 | \pmcreated{2013-03-22 12:31:19} 5 | \pmmodified{2013-03-22 12:31:19} 6 | \pmowner{mps}{409} 7 | \pmmodifier{mps}{409} 8 | \pmtitle{size (of a graph)} 9 | \pmrecord{8}{32761} 10 | \pmprivacy{1} 11 | \pmauthor{mps}{409} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmsynonym{size}{SizeofAGraph} 16 | \pmrelated{Graph} 17 | \pmrelated{OrderOfAGraph} 18 | \pmrelated{MantelsTheorem} 19 | 20 | \endmetadata 21 | 22 | % this is the default PlanetMath preamble. as your knowledge 23 | % of TeX increases, you will probably want to edit this, but 24 | % it should be fine as is for beginners. 25 | 26 | % almost certainly you want these 27 | \usepackage{amssymb} 28 | \usepackage{amsmath} 29 | \usepackage{amsfonts} 30 | 31 | % used for TeXing text within eps files 32 | %\usepackage{psfrag} 33 | % need this for including graphics (\includegraphics) 34 | %\usepackage{graphicx} 35 | % for neatly defining theorems and propositions 36 | %\usepackage{amsthm} 37 | % making logically defined graphics 38 | %%%\usepackage{xypic} 39 | 40 | % there are many more packages, add them here as you need them 41 | 42 | % define commands here 43 | \begin{document} 44 | The \emph{size} of a graph $G$ is the number of edges in $G$; it is denoted by $e(G)$. $G(n,m)$ denotes an \emph{arbitrary graph of order n and size m}. 45 | 46 | 47 | \footnotesize{Adapted with permission of the author from \emph{\PMlinkescapetext{Modern Graph Theory}} by B\'{e}la Bollob\'{a}s, published by Springer-Verlag New York, Inc., 1998.} 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /05C99-MinorofAGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{MinorofAGraph} 4 | \pmcreated{2013-03-22 12:31:46} 5 | \pmmodified{2013-03-22 12:31:46} 6 | \pmowner{digitalis}{76} 7 | \pmmodifier{digitalis}{76} 8 | \pmtitle{minor (of a graph)} 9 | \pmrecord{4}{32770} 10 | \pmprivacy{1} 11 | \pmauthor{digitalis}{76} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmsynonym{minor}{MinorofAGraph} 16 | %\pmkeywords{contraction} 17 | \pmrelated{GraphMinorTheorem} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | \begin{document} 43 | A graph $H$ is a \emph{minor} of $G$, written $G \succ H$ or $H \prec G$, if it is a subgraph of a graph obtained from $G$ by a sequence of edge-contractions. 44 | 45 | 46 | \footnotesize{Adapted with permission of the author from \emph{\PMlinkescapetext{Modern Graph Theory}} by B\'{e}la Bollob\'{a}s, published by Springer-Verlag New York, Inc., 1998.} 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /05A30-GaussianPolynomials.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{GaussianPolynomials} 4 | \pmcreated{2013-03-22 11:49:49} 5 | \pmmodified{2013-03-22 11:49:49} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{Gaussian polynomials} 9 | \pmrecord{10}{30378} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A30} 15 | \pmclassification{msc}{05A10} 16 | \pmclassification{msc}{16S36} 17 | \pmclassification{msc}{26A09} 18 | \pmclassification{msc}{26A18} 19 | \pmclassification{msc}{15A04} 20 | \pmsynonym{q-binomial coefficients}{GaussianPolynomials} 21 | \pmrelated{ContentOfPolynomial} 22 | 23 | \endmetadata 24 | 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | \usepackage{graphicx} 29 | %%%%\usepackage{xypic} 30 | \begin{document} 31 | For an indeterminate $q$ and integers $n \ge m \ge 0$ 32 | we define the following: \par 33 | (a) $(m)_q = q^{m-1} + q^{m-2} + \cdots + 1$ for $m>0$, \par 34 | (b) $(m!)_q = (m)_q (m-1)_q \cdots (1)_q$ for $m>0$, and $(0!)_q = 1$, \par 35 | (c) ${n \choose m}_q = \frac{(n!)_q}{(m!)_q ((n-m)!)_q}$. 36 | If $m>n$ then we define ${n \choose m}_q=0$. \par 37 | The expressions ${n \choose m}_q$ are called 38 | {\it $q$-binomial coefficients} or {\it Gaussian polynomials}.\par 39 | Note: if we replace $q$ with 1, then we obtain the familiar integers, factorials, and binomial coefficients. Specifically,\par 40 | (a) $(m)_1 = m$, \par 41 | (b) $(m!)_1 = m!$, \par 42 | (c) ${n \choose m}_1 = {n \choose m}$.\par 43 | (d) ${m \choose m}_q=1$. 44 | %%%%% 45 | %%%%% 46 | %%%%% 47 | %%%%% 48 | \end{document} 49 | -------------------------------------------------------------------------------- /05C05-ExtendedBinaryTree.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ExtendedBinaryTree} 4 | \pmcreated{2013-03-22 12:31:36} 5 | \pmmodified{2013-03-22 12:31:36} 6 | \pmowner{aoh45}{5079} 7 | \pmmodifier{aoh45}{5079} 8 | \pmtitle{extended binary tree} 9 | \pmrecord{8}{32766} 10 | \pmprivacy{1} 11 | \pmauthor{aoh45}{5079} 12 | \pmtype{Data Structure} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmrelated{BinaryTree} 16 | \pmrelated{CompleteBinaryTree} 17 | \pmrelated{ExternalPathLength} 18 | \pmrelated{WeightedPathLength} 19 | \pmrelated{MinimumWeightedPathLength} 20 | \pmdefines{external node} 21 | \pmdefines{internal node} 22 | 23 | \endmetadata 24 | 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | \usepackage{graphicx} 29 | \begin{document} 30 | An \emph{extended binary tree} is a transformation of any binary tree into a complete binary tree. This transformation consists of replacing every null subtree of the original tree with ``special nodes.'' The nodes from the original tree are then \emph{internal nodes}, while the ``special nodes'' are \emph{external nodes}. 31 | 32 | For instance, consider the following binary tree. 33 | 34 | \begin{center} 35 | \includegraphics{tree1} 36 | \end{center} 37 | 38 | The following tree is its extended binary tree. Empty circles represent internal nodes, and filled circles represent external nodes. 39 | 40 | \begin{center} 41 | \includegraphics{tree2} 42 | \end{center} 43 | 44 | Every internal node in the extended tree has exactly two children, and every external node is a leaf. The result is a complete binary tree. 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05C70-EdgeCovering.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{EdgeCovering} 4 | \pmcreated{2013-03-22 12:40:02} 5 | \pmmodified{2013-03-22 12:40:02} 6 | \pmowner{vampyr}{22} 7 | \pmmodifier{vampyr}{22} 8 | \pmtitle{edge covering} 9 | \pmrecord{8}{32940} 10 | \pmprivacy{1} 11 | \pmauthor{vampyr}{22} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C70} 15 | \pmrelated{Matching} 16 | \pmdefines{minimal edge covering} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | Let $G$ be a graph. An \emph{edge covering} $C$ on $G$ is a subset of the vertices of $G$ such that each edge in $G$ is incident with at least one vertex in $C$. 43 | 44 | For any graph, the \PMlinkescapeword{entire} vertex set is a trivial edge covering. Generally, we are more interested in \emph{minimal coverings}. A minimal edge covering is simply an edge covering of the least possible \PMlinkescapeword{size}. 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05C99-Homeomorphism1.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Homeomorphism1} 4 | \pmcreated{2013-03-22 12:31:55} 5 | \pmmodified{2013-03-22 12:31:55} 6 | \pmowner{digitalis}{76} 7 | \pmmodifier{digitalis}{76} 8 | \pmtitle{homeomorphism} 9 | \pmrecord{6}{32773} 10 | \pmprivacy{1} 11 | \pmauthor{digitalis}{76} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmrelated{Subdivision} 16 | \pmrelated{Realization} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | We say that a graph $G$ is \emph{homeomorphic} to graph $H$ if the 43 | realization $R(G)$ of $G$ is topologically \PMlinkname{homeomorphic}{Homeomorphism} to $R(H)$ or, equivalently, $G$ and $H$ have isomorphic subdivisions. 44 | 45 | 46 | \footnotesize{Adapted with permission of the author from \emph{\PMlinkescapetext{Modern Graph Theory}} by B\'{e}la Bollob\'{a}s, published by Springer-Verlag New York, Inc., 1998.} 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /05C05-ExternalPathLength.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ExternalPathLength} 4 | \pmcreated{2013-03-22 12:32:03} 5 | \pmmodified{2013-03-22 12:32:03} 6 | \pmowner{Logan}{6} 7 | \pmmodifier{Logan}{6} 8 | \pmtitle{external path length} 9 | \pmrecord{4}{32776} 10 | \pmprivacy{1} 11 | \pmauthor{Logan}{6} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmrelated{ExtendedBinaryTree} 16 | \pmrelated{WeightedPathLength} 17 | \pmdefines{external path length} 18 | \pmdefines{internal path length} 19 | 20 | \endmetadata 21 | 22 | \usepackage{amssymb} 23 | \usepackage{amsmath} 24 | \usepackage{amsfonts} 25 | \usepackage{graphicx} 26 | \begin{document} 27 | Given a binary tree $T$, construct its extended binary tree $T'$. 28 | The \emph{external path length} of $T$ is then defined to be the sum of the lengths of the paths to each of the external nodes. 29 | 30 | For example, let $T$ be the following tree. 31 | 32 | \begin{center} 33 | \includegraphics{tree.3} 34 | \end{center} 35 | 36 | The extended binary tree of $T$ is 37 | 38 | \begin{center} 39 | \includegraphics{tree.4} 40 | \end{center} 41 | 42 | The external path length of $T$ (denoted $E$) is 43 | 44 | $$ 45 | E = 2 + 3 + 3 + 3 + 3 + 3 + 3 = 20 46 | $$ 47 | 48 | The \emph{internal path length} of $T$ is defined to be the sum of the lengths of the paths to each of the internal nodes. The internal path length of our example tree (denoted $I$) is 49 | 50 | $$ 51 | I = 1 + 2 + 0 + 2 + 1 + 2 = 8 52 | $$ 53 | 54 | Note that in this case $E = I + 2n$, where $n$ is the number of internal nodes. This happens to hold for all binary trees. 55 | %%%%% 56 | %%%%% 57 | \end{document} 58 | -------------------------------------------------------------------------------- /05C25-HyperbolicGroup.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{HyperbolicGroup} 4 | \pmcreated{2013-03-22 17:11:43} 5 | \pmmodified{2013-03-22 17:11:43} 6 | \pmowner{Wkbj79}{1863} 7 | \pmmodifier{Wkbj79}{1863} 8 | \pmtitle{hyperbolic group} 9 | \pmrecord{6}{39514} 10 | \pmprivacy{1} 11 | \pmauthor{Wkbj79}{1863} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C25} 15 | \pmclassification{msc}{20F06} 16 | \pmclassification{msc}{54E35} 17 | \pmsynonym{hyperbolicity}{HyperbolicGroup} 18 | \pmrelated{RealTree} 19 | 20 | \endmetadata 21 | 22 | \usepackage{amssymb} 23 | \usepackage{amsmath} 24 | \usepackage{amsfonts} 25 | \usepackage{pstricks} 26 | \usepackage{psfrag} 27 | \usepackage{graphicx} 28 | \usepackage{amsthm} 29 | %%\usepackage{xypic} 30 | 31 | \begin{document} 32 | A finitely generated group $G$ is \emph{hyperbolic} if, for some finite set of generators $A$ of $G$, the Cayley graph $\Gamma(G,A)$, considered as a metric space with $d(x,y)$ being the minimum number of edges one must traverse to get from $x$ to $y$, is a hyperbolic metric space. 33 | 34 | Hyperbolicity is a group-theoretic property. That is, if $A$ and $B$ are finite sets of generators of a group $G$ and $\Gamma(G,A)$ is a hyperbolic metric space, then $\Gamma(G,B)$ is a hyperbolic metric space. 35 | 36 | \PMlinkescapetext{Simple} examples of hyperbolic groups include finite groups and free groups. If $G$ is a finite group, then for any $x,y \in G$, we have that $d(x,y) \le |G|$. (See the entry \PMlinkname{Cayley graph of $S_3$}{CayleyGraphOfS_3} for a pictorial example.) If $G$ is a free group, then its Cayley graph is a real tree. 37 | %%%%% 38 | %%%%% 39 | \end{document} 40 | -------------------------------------------------------------------------------- /05C99-EulersPolyhedronTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{EulersPolyhedronTheorem} 4 | \pmcreated{2013-03-22 12:25:28} 5 | \pmmodified{2013-03-22 12:25:28} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{Euler's polyhedron theorem} 9 | \pmrecord{6}{32433} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmsynonym{Euler's formula}{EulersPolyhedronTheorem} 16 | %\pmkeywords{graph} 17 | %\pmkeywords{vertex} 18 | %\pmkeywords{edge} 19 | %\pmkeywords{face} 20 | %\pmkeywords{vertices} 21 | \pmrelated{Graph} 22 | \pmrelated{GraphTheory} 23 | \pmrelated{PoincareFormula} 24 | \pmrelated{Polytope} 25 | 26 | \endmetadata 27 | 28 | % this is the default PlanetMath preamble. as your knowledge 29 | % of TeX increases, you will probably want to edit this, but 30 | % it should be fine as is for beginners. 31 | 32 | % almost certainly you want these 33 | \usepackage{amssymb} 34 | \usepackage{amsmath} 35 | \usepackage{amsfonts} 36 | 37 | % used for TeXing text within eps files 38 | %\usepackage{psfrag} 39 | % need this for including graphics (\includegraphics) 40 | %\usepackage{graphicx} 41 | % for neatly defining theorems and propositions 42 | %\usepackage{amsthm} 43 | % making logically defined graphics 44 | %%%\usepackage{xypic} 45 | 46 | % there are many more packages, add them here as you need them 47 | 48 | % define commands here 49 | \begin{document} 50 | \newtheorem{thm}{Theorem} 51 | \begin{thm}[] 52 | If a connected plane graph has $n$ vertices, $m$ edges, and $f$ faces, then 53 | $$n - m + f = 2.$$ 54 | \end{thm} 55 | %%%%% 56 | %%%%% 57 | \end{document} 58 | -------------------------------------------------------------------------------- /05A10-UpperAndLowerBoundsToBinomialCoefficient.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{UpperAndLowerBoundsToBinomialCoefficient} 4 | \pmcreated{2013-03-22 13:29:53} 5 | \pmmodified{2013-03-22 13:29:53} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{upper and lower bounds to binomial coefficient} 9 | \pmrecord{6}{34074} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A10} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | Given two integers $n,k>0$ such that $k\le n$, we have the following inequalities for the binomial coefficient ${n\choose k}$: 41 | \begin{eqnarray*} 42 | {n \choose k} & \le & \frac{n^k}{k!} \\ 43 | {n \choose k} & \le & \left(\frac{n\cdot e}{k}\right)^k \\ 44 | {n \choose k} & \ge & \left(\frac{n}{k}\right)^k \\ 45 | \end{eqnarray*} 46 | Here $e$ is the base of natural logarithms. 47 | Also, for large $n$, ${n \choose k} \approx \frac{n^k}{k!}$. 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /05C83-EquivalenceBetweenTheMinorAndTopologicalMinorOfK5OrK331.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{EquivalenceBetweenTheMinorAndTopologicalMinorOfK5OrK33} 4 | \pmcreated{2013-03-22 17:47:13} 5 | \pmmodified{2013-03-22 17:47:13} 6 | \pmowner{jwaixs}{18148} 7 | \pmmodifier{jwaixs}{18148} 8 | \pmtitle{equivalence between the minor and topological minor of $K_5$ or $K_{3,3}$} 9 | \pmrecord{8}{40247} 10 | \pmprivacy{1} 11 | \pmauthor{jwaixs}{18148} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C83} 15 | \pmclassification{msc}{05C10} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \begin{document} 42 | A graph $ G $ contains $ K_5 $ or $ K_{3,3} $ as a minor iff it contains $ K_5 $ or $ K_{3,3} $ as a \PMlinkname{topological minor}{subdivision}. Where $ K_5$ is the complete graph of order 5 and $ K_{3,3}$ is the complete bipartite graph of order 6. 43 | 44 | Remark that this theorem shows that Wagner's theorem and Kuratowski's theorem are equivalent. 45 | %%%%% 46 | %%%%% 47 | \end{document} 48 | -------------------------------------------------------------------------------- /05A10-DoubleFactorial.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{DoubleFactorial} 4 | \pmcreated{2013-03-22 12:24:54} 5 | \pmmodified{2013-03-22 12:24:54} 6 | \pmowner{drini}{3} 7 | \pmmodifier{drini}{3} 8 | \pmtitle{double factorial} 9 | \pmrecord{9}{32318} 10 | \pmprivacy{1} 11 | \pmauthor{drini}{3} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A10} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | \begin{document} 40 | The \emph{double factorial} of a positive integer $n$ is the product $n!!$ of the positive integers less than or equal to $n$ that have the same parity as $n$, that is, 41 | \[n!! = n (n-2) (n-4)\cdots k_n\] 42 | where $k_n$ denotes $1$ if $n$ is an odd number and $2$ if $n$ is an even number. 43 | 44 | For example, 45 | \[ 7!! = 7 \cdot 5 \cdot 3 \cdot 1 = 105 \] 46 | \[ 10!! = 10\cdot 8\cdot 6\cdot 4\cdot 2 = 3840 \] 47 | 48 | Note that $n!!$ is not the same as $(n!)!$. 49 | 50 | Observe that $(2n)!! = 2^n n!$ and $(2n+1)!! = \frac{(2n+1)!}{2^n n!}$. 51 | %%%%% 52 | %%%%% 53 | \end{document} 54 | -------------------------------------------------------------------------------- /05C38-EulerPath.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{EulerPath} 4 | \pmcreated{2013-03-22 12:02:04} 5 | \pmmodified{2013-03-22 12:02:04} 6 | \pmowner{CWoo}{3771} 7 | \pmmodifier{CWoo}{3771} 8 | \pmtitle{Euler path} 9 | \pmrecord{18}{31043} 10 | \pmprivacy{1} 11 | \pmauthor{CWoo}{3771} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C38} 15 | %\pmkeywords{Euler path} 16 | %\pmkeywords{Euler circuit} 17 | \pmrelated{EulerCircuit} 18 | \pmrelated{Graph} 19 | \pmdefines{Euler cycle} 20 | 21 | \endmetadata 22 | 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | \usepackage{graphicx} 27 | %%%\usepackage{xypic} 28 | \begin{document} 29 | An \emph{Euler path} in a graph is a path which traverses each edge of the graph exactly once. An Euler path which is a cycle is called an \emph{Euler cycle}. For loopless graphs without isolated vertices, the existence of an Euler path implies the connectedness of the graph, since traversing every edge of such a graph requires visiting each vertex at least once. 30 | 31 | If a connected graph has an Euler path, one can be constructed by applying Fleury's algorithm. A connected graph has an Euler path if it has exactly zero or two vertices of odd degree. If every vertex has even degree, the graph has an Euler cycle. 32 | 33 | \begin{center} 34 | \includegraphics[width=1in,height=1in]{ecircuit} 35 | \end{center} 36 | 37 | This graph has an Euler cycle. All of its vertices are of even degree. 38 | 39 | \begin{center} 40 | \includegraphics[width=1in,height=1in]{epath} 41 | \end{center} 42 | 43 | This graph has an Euler path which is not a cycle. It has exactly two vertices of odd degree. 44 | 45 | %%%%% 46 | %%%%% 47 | %%%%% 48 | \end{document} 49 | -------------------------------------------------------------------------------- /05C50-AlonChungLemma.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{AlonChungLemma} 4 | \pmcreated{2013-03-22 17:26:47} 5 | \pmmodified{2013-03-22 17:26:47} 6 | \pmowner{kshum}{5987} 7 | \pmmodifier{kshum}{5987} 8 | \pmtitle{Alon-Chung lemma} 9 | \pmrecord{6}{39827} 10 | \pmprivacy{1} 11 | \pmauthor{kshum}{5987} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C50} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | 27 | % used for TeXing text within eps files 28 | %\usepackage{psfrag} 29 | % need this for including graphics (\includegraphics) 30 | %\usepackage{graphicx} 31 | % for neatly defining theorems and propositions 32 | %\usepackage{amsthm} 33 | % making logically defined graphics 34 | %%%\usepackage{xypic} 35 | 36 | % there are many more packages, add them here as you need them 37 | 38 | % define commands here 39 | 40 | \begin{document} 41 | Let $G= (V,E)$ be a undirected graph of $n$ vertices such that the degree of each vertex is equal to $d$. Let $X$ be a subset of $V$. Then the number of edges in the subgraph induced by $X$ is at most 42 | \[ 43 | \frac{1}{2n} \Big( d|X|^2 + \lambda |X|(n- |X|) \Big) 44 | \] 45 | where $\lambda$ is the second largest eigenvalue of the adjacency matrix of $G$. 46 | 47 | 48 | \bigskip 49 | 50 | 51 | {\bf Reference:} N. Alon and F. R. K. Chung, ``Explicit construction of linear sized 52 | tolerant networks,'' Discrete Math., vol. 72, pp. 15-19, 1988. 53 | 54 | %%%%% 55 | %%%%% 56 | \end{document} 57 | -------------------------------------------------------------------------------- /05D10-Coloring.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Coloring} 4 | \pmcreated{2013-03-22 12:55:43} 5 | \pmmodified{2013-03-22 12:55:43} 6 | \pmowner{Henry}{455} 7 | \pmmodifier{Henry}{455} 8 | \pmtitle{coloring} 9 | \pmrecord{5}{33283} 10 | \pmprivacy{1} 11 | \pmauthor{Henry}{455} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05D10} 15 | \pmsynonym{colouring}{Coloring} 16 | \pmrelated{Partition} 17 | \pmrelated{GraphTheory} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | %\PMlinkescapeword{theory} 43 | \begin{document} 44 | A \emph{coloring} of a set $X$ by $Y$ is just a function $f:X\rightarrow Y$. The term coloring is used because the function can be thought of as assigning a ``color'' from $Y$ to each element of $X$. 45 | 46 | Any coloring provides a partition of $X$: for each $y\in Y$, $f^{-1}(y)$, the set of elements $x$ such that $f(x)=y$, is one element of the partition. Since $f$ is a function, the sets in the partition are disjoint, and since it is a total function, their union is $X$. 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /05C99-Edgecontraction.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Edgecontraction} 4 | \pmcreated{2013-03-22 12:31:43} 5 | \pmmodified{2013-03-22 12:31:43} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{edge-contraction} 9 | \pmrecord{5}{32769} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmrelated{TheoremOn3ConnectedGraphs} 16 | \pmdefines{contraction} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | Given an edge $xy$ of a graph $G$, the graph $G/xy$ is obtained from $G$ by \emph{contracting} the edge $xy$; that is, to get $G/xy$ we identify the vertices $x$ and $y$ and remove all loops and duplicate edges. A graph $G'$ obtained by a sequence of edge-contractions is said to be a \emph{contraction} of $G$. 43 | 44 | 45 | \footnotesize{Adapted with permission of the author from \emph{\PMlinkescapetext{Modern Graph Theory}} by B\'{e}la Bollob\'{a}s, published by Springer-Verlag New York, Inc., 1998.} 46 | %%%%% 47 | %%%%% 48 | \end{document} 49 | -------------------------------------------------------------------------------- /pdf/nocando: -------------------------------------------------------------------------------- 1 | 05A10-CatalanNumbers 2 | 05A10-PascalsTriangle 3 | 05A10-ThingsCountedByTheCatalanNumbers 4 | 05A15-EnumerationOfLatticeWalks 5 | 05A15-LatticePathsAndBallotNumbers 6 | 05A17-IntegerPartition 7 | 05B25-DeBruijnErdHosTheorem 8 | 05B50-Polyomino 9 | 05C05-ChildNodeofATree 10 | 05C05-ExtendedBinaryTree 11 | 05C05-ExternalPathLength 12 | 05C05-InternalNodeofATree 13 | 05C05-LeafNodeofATree 14 | 05C05-MinimumWeightedPathLength 15 | 05C05-ParentNodeinATree 16 | 05C05-RootedTree 17 | 05C05-RootofATree 18 | 05C05-WeightedPathLength 19 | 05C10-ExampleOfPlanarGraphWithTwoDifferentEmbeddingsIntoThePlane 20 | 05C10-FourcolorConjecture 21 | 05C10-PlanarGraph 22 | 05C15-BipartiteGraph 23 | 05C15-ChromaticNumberOfAMetricSpace 24 | 05C15-ChromaticNumber 25 | 05C15-ColouringProblem 26 | 05C15-CompleteBipartiteGraph 27 | 05C15-CompleteKpartiteGraph 28 | 05C15-HeawoodNumber 29 | 05C15-VizingsTheorem 30 | 05C20-KautzGraph 31 | 05C25-HyperbolicGroup 32 | 05C30-EnumeratingGraphs 33 | 05C38-BridgesOfKonigsberg 34 | 05C38-Cycle 35 | 05C38-EulerPath 36 | 05C38-Path1 37 | 05C45-EulerCircuit 38 | 05C45-PetersenGraph 39 | 05C50-AdjacencyMatrix 40 | 05C60-GraphIsomorphism 41 | 05C69-Clique 42 | 05C70-BipartiteMatching 43 | 05C75-GraphHomomorphism 44 | 05C75-LineGraph 45 | 05C75-MooreGraph 46 | 05C80-ProbabilisticMethod 47 | 05C83-EquivalenceBetweenTheMinorAndTopologicalMinorOfK5OrK33 48 | 05C85-TheAnnalsOfHavingCompensatedToCompleteSurveys 49 | 05C85-TheRealHistoryOfGettingSettledToAccomplishStudies 50 | 05C90-HasseDiagram 51 | 05C99-CompleteGraph 52 | 05C99-EulersPolyhedronTheoremProofOf 53 | 05C99-GraphMinorTheorem 54 | 05C99-Graph 55 | 05C99-KneserGraphs 56 | 05C99-LocallyFiniteGraph 57 | 05C99-WheelGraph 58 | 05D05-LYMInequality 59 | 05D05-SpernersTheorem 60 | 05D10-BehrendsConstruction 61 | 05D40-RamseysTheorem 62 | 05E99-StarProduct 63 | -------------------------------------------------------------------------------- /05C05-Antichain.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Antichain} 4 | \pmcreated{2013-03-22 12:52:25} 5 | \pmmodified{2013-03-22 12:52:25} 6 | \pmowner{Henry}{455} 7 | \pmmodifier{Henry}{455} 8 | \pmtitle{antichain} 9 | \pmrecord{6}{33212} 10 | \pmprivacy{1} 11 | \pmauthor{Henry}{455} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C05} 15 | \pmclassification{msc}{03E05} 16 | \pmrelated{TreeSetTheoretic} 17 | \pmrelated{Aronszajn} 18 | \pmdefines{antichain} 19 | \pmdefines{maximal antichain} 20 | 21 | \endmetadata 22 | 23 | % this is the default PlanetMath preamble. as your knowledge 24 | % of TeX increases, you will probably want to edit this, but 25 | % it should be fine as is for beginners. 26 | 27 | % almost certainly you want these 28 | \usepackage{amssymb} 29 | \usepackage{amsmath} 30 | \usepackage{amsfonts} 31 | 32 | % used for TeXing text within eps files 33 | %\usepackage{psfrag} 34 | % need this for including graphics (\includegraphics) 35 | %\usepackage{graphicx} 36 | % for neatly defining theorems and propositions 37 | %\usepackage{amsthm} 38 | % making logically defined graphics 39 | %%%\usepackage{xypic} 40 | 41 | % there are many more packages, add them here as you need them 42 | 43 | % define commands here 44 | %\PMlinkescapeword{theory} 45 | \begin{document} 46 | A subset $A$ of a poset $(P,<_P)$ is an \emph{antichain} if no two elements are comparable. That is, if $a,b\in A$ then $a\nless_P b$ and $b\nless_P a$. 47 | 48 | A \emph{maximal antichain} of $T$ is one which is maximal. 49 | 50 | In particular, if $(P,<_P)$ is a tree then the maximal antichains are exactly those antichains which intersect every branch, and if the tree is splitting then every level is a maximal antichain. 51 | %%%%% 52 | %%%%% 53 | \end{document} 54 | -------------------------------------------------------------------------------- /05C69-SizeOfMaximalIndependentSetAndChromaticNumber.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SizeOfMaximalIndependentSetAndChromaticNumber} 4 | \pmcreated{2013-03-22 14:30:02} 5 | \pmmodified{2013-03-22 14:30:02} 6 | \pmowner{kshum}{5987} 7 | \pmmodifier{kshum}{5987} 8 | \pmtitle{size of maximal independent set and chromatic number} 9 | \pmrecord{10}{36037} 10 | \pmprivacy{1} 11 | \pmauthor{kshum}{5987} 12 | \pmtype{Theorem} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C69} 15 | \pmclassification{msc}{05C15} 16 | %\pmkeywords{chromatic number} 17 | %\pmkeywords{independent sets} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | %\usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | \usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | \begin{document} 43 | Let $\alpha(G)$ be the size of the largest independent set in a graph $G$, and $\chi(G)$ the chromatic number of $G$. 44 | 45 | {\bf Theorem:} $\alpha(G)\chi(G) \geq |G|$. 46 | 47 | \bigskip 48 | 49 | \begin{proof} 50 | The vertices of $G$ can be partitioned into $\chi(G)$ monochromatic classes. Each class is an independent set, and hence cannot have size larger than $\alpha(G)$. 51 | \end{proof} 52 | %%%%% 53 | %%%%% 54 | \end{document} 55 | -------------------------------------------------------------------------------- /05A10-SomeFormulasInvolvingRisingFactorial.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SomeFormulasInvolvingRisingFactorial} 4 | \pmcreated{2013-03-22 17:49:12} 5 | \pmmodified{2013-03-22 17:49:12} 6 | \pmowner{Wkbj79}{1863} 7 | \pmmodifier{Wkbj79}{1863} 8 | \pmtitle{some formulas involving rising factorial} 9 | \pmrecord{4}{40283} 10 | \pmprivacy{1} 11 | \pmauthor{Wkbj79}{1863} 12 | \pmtype{Result} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A10} 15 | 16 | \endmetadata 17 | 18 | \usepackage{amssymb} 19 | \usepackage{amsmath} 20 | \usepackage{amsfonts} 21 | \usepackage{pstricks} 22 | \usepackage{psfrag} 23 | \usepackage{graphicx} 24 | \usepackage{amsthm} 25 | %%\usepackage{xypic} 26 | 27 | \begin{document} 28 | \PMlinkescapeword{formula} 29 | \PMlinkescapeword{relation} 30 | 31 | Recall that, for $a\in\mathbb{C}$ and $n$ a nonnegative integer, the rising factorial $(a)_n$ is defined by 32 | \[ 33 | (a)_n=\prod_{k=0}^{n-1}(a+k). 34 | \] 35 | 36 | The following results hold regarding the rising factorial: 37 | 38 | \begin{itemize} 39 | \item For all $a\in\mathbb{C}$, we have $(a)_0=1$. 40 | \item For all nonnegative integers $n$, we have $(1)_n=n!$. 41 | \item The binomial coefficients are given by 42 | \[ 43 | \binom{a}{n}=\frac{(-1)^n(-a)_n}{n!}. 44 | \] 45 | \item The rising factorial relates to the gamma function. One relation is given by the formula 46 | \[ 47 | (a)_n=\frac{\Gamma(a+n)}{\Gamma(a)}. 48 | \] 49 | This formula can be used to extend the definition of rising factorial so that $n$ can be any complex number provided that $a+n$ is not a nonpositive integer. 50 | \item Another relation between the rising factorial and the gamma function is given by 51 | \[ 52 | \Gamma(a)=\lim_{n\to\infty} \frac{n!\,n^{a-1}}{(a)_n}. 53 | \] 54 | \end{itemize} 55 | %%%%% 56 | %%%%% 57 | \end{document} 58 | -------------------------------------------------------------------------------- /05C99-Block.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Block} 4 | \pmcreated{2013-03-22 12:32:00} 5 | \pmmodified{2013-03-22 12:32:00} 6 | \pmowner{digitalis}{76} 7 | \pmmodifier{digitalis}{76} 8 | \pmtitle{block} 9 | \pmrecord{4}{32775} 10 | \pmprivacy{1} 11 | \pmauthor{digitalis}{76} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmrelated{Cutvertex} 16 | \pmrelated{Bridge} 17 | 18 | \endmetadata 19 | 20 | % this is the default PlanetMath preamble. as your knowledge 21 | % of TeX increases, you will probably want to edit this, but 22 | % it should be fine as is for beginners. 23 | 24 | % almost certainly you want these 25 | \usepackage{amssymb} 26 | \usepackage{amsmath} 27 | \usepackage{amsfonts} 28 | 29 | % used for TeXing text within eps files 30 | %\usepackage{psfrag} 31 | % need this for including graphics (\includegraphics) 32 | %\usepackage{graphicx} 33 | % for neatly defining theorems and propositions 34 | %\usepackage{amsthm} 35 | % making logically defined graphics 36 | %%%\usepackage{xypic} 37 | 38 | % there are many more packages, add them here as you need them 39 | 40 | % define commands here 41 | \begin{document} 42 | A subgraph $B$ of a graph $G$ is a \emph{block of} $G$ if either it is a bridge (together with the vertices incident with the bridge) or else it is a maximal 2-connected subgraph of $G$. 43 | 44 | Any two blocks of a graph $G$ have at most one vertex in common. Also, every vertex belonging to at least two blocks is a cutvertex of $G$, and, conversely, every cutvertex belongs to at least two blocks. 45 | 46 | 47 | \footnotesize{Adapted with permission of the author from \emph{\PMlinkescapetext{Modern Graph Theory}} by B\'{e}la Bollob\'{a}s, published by Springer-Verlag New York, Inc., 1998.} 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /05A10-ExponentialFactorial.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ExponentialFactorial} 4 | \pmcreated{2013-03-22 16:01:38} 5 | \pmmodified{2013-03-22 16:01:38} 6 | \pmowner{CompositeFan}{12809} 7 | \pmmodifier{CompositeFan}{12809} 8 | \pmtitle{exponential factorial} 9 | \pmrecord{7}{38068} 10 | \pmprivacy{1} 11 | \pmauthor{CompositeFan}{12809} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A10} 15 | \pmrelated{Factorial} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \begin{document} 42 | Given a positive integer $n$, the "power tower" $n^{(n - 1)^{(n - 2) \dots }}$ is the {\em exponential factorial} of $n$. The recurrence relation is $a_1 = 1$, $a_n = n^{a_{n - 1}}$ for $n > 1$. 43 | 44 | So for example, $9 = 3^{2^1}$, $262144 = 4^{3^{2^1}}$. The exponential factorial for 5 has almost two hundred thousand base 10 digits. The ones that are small enough are listed in sequence A049384 of Sloane's OEIS. 45 | 46 | The sum of the reciprocals of the exponential factorials is a Liouville number. $$\sum_{i = 1}^\infty {1 \over a_i} \approx 1.6111149258083767361111111$$ 47 | %%%%% 48 | %%%%% 49 | \end{document} 50 | -------------------------------------------------------------------------------- /05C99-OrderofAGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{OrderofAGraph} 4 | \pmcreated{2013-03-22 12:31:23} 5 | \pmmodified{2013-03-22 12:31:23} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{order (of a graph)} 9 | \pmrecord{8}{32762} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmsynonym{order}{OrderofAGraph} 16 | \pmrelated{Graph} 17 | \pmrelated{SizeOfAGraph} 18 | \pmrelated{MantelsTheorem} 19 | 20 | \endmetadata 21 | 22 | % this is the default PlanetMath preamble. as your knowledge 23 | % of TeX increases, you will probably want to edit this, but 24 | % it should be fine as is for beginners. 25 | 26 | % almost certainly you want these 27 | \usepackage{amssymb} 28 | \usepackage{amsmath} 29 | \usepackage{amsfonts} 30 | 31 | % used for TeXing text within eps files 32 | %\usepackage{psfrag} 33 | % need this for including graphics (\includegraphics) 34 | %\usepackage{graphicx} 35 | % for neatly defining theorems and propositions 36 | %\usepackage{amsthm} 37 | % making logically defined graphics 38 | %%%\usepackage{xypic} 39 | 40 | % there are many more packages, add them here as you need them 41 | 42 | % define commands here 43 | \begin{document} 44 | The \emph{order} of a graph $G$ is the number of vertices in $G$; it is denoted by $|G|$. The same notation is used for the number of elements (cardinality) of a set. Thus, $|G| = |V(G)|$. We write $G^n$ for an \emph{arbitrary graph of order n}. Similarly, $G(n,m)$ denotes an \emph{arbitrary graph of order n and size m}. 45 | 46 | 47 | \footnotesize{Adapted with permission of the author from \emph{\PMlinkescapetext{Modern Graph Theory}} by B\'{e}la Bollob\'{a}s, published by Springer-Verlag New York, Inc., 1998.} 48 | %%%%% 49 | %%%%% 50 | \end{document} 51 | -------------------------------------------------------------------------------- /05C99-CompleteGraph.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{CompleteGraph} 4 | \pmcreated{2013-03-22 12:16:57} 5 | \pmmodified{2013-03-22 12:16:57} 6 | \pmowner{vampyr}{22} 7 | \pmmodifier{vampyr}{22} 8 | \pmtitle{complete graph} 9 | \pmrecord{10}{31757} 10 | \pmprivacy{1} 11 | \pmauthor{vampyr}{22} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C99} 15 | \pmsynonym{complete}{CompleteGraph} 16 | \pmsynonym{clique}{CompleteGraph} 17 | \pmrelated{Tournament} 18 | 19 | \endmetadata 20 | 21 | % this is the default PlanetMath preamble. as your knowledge 22 | % of TeX increases, you will probably want to edit this, but 23 | % it should be fine as is for beginners. 24 | 25 | % almost certainly you want these 26 | \usepackage{amssymb} 27 | \usepackage{amsmath} 28 | \usepackage{amsfonts} 29 | 30 | % used for TeXing text within eps files 31 | %\usepackage{psfrag} 32 | % need this for including graphics (\includegraphics) 33 | \usepackage{graphicx} 34 | % for neatly defining theorems and propositions 35 | %\usepackage{amsthm} 36 | % making logically defined graphics 37 | %%%%\usepackage{xypic} 38 | 39 | % there are many more packages, add them here as you need them 40 | 41 | % define commands here 42 | \begin{document} 43 | The \emph{complete graph} with $n$ vertices, denoted $K_n$, contains all possible edges; that is, any two vertices are adjacent. 44 | 45 | The complete graph of $4$ vertices, or $K_4$ looks like this: 46 | 47 | \begin{center} 48 | \includegraphics[scale=1.0]{k4.eps} 49 | \end{center} 50 | 51 | The number of edges in $K_n$ is the $n-1$th triangular number. Every vertex in $K_n$ has degree $n-1$; therefore $K_n$ has an Euler circuit if and only if $n$ is odd. A complete graph always has a Hamiltonian path, and the chromatic number of $K_n$ is always $n$. 52 | %%%%% 53 | %%%%% 54 | %%%%% 55 | \end{document} 56 | -------------------------------------------------------------------------------- /05C38-SimplePath.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{SimplePath} 4 | \pmcreated{2013-03-22 12:30:42} 5 | \pmmodified{2013-03-22 12:30:42} 6 | \pmowner{mps}{409} 7 | \pmmodifier{mps}{409} 8 | \pmtitle{simple path} 9 | \pmrecord{6}{32747} 10 | \pmprivacy{1} 11 | \pmauthor{mps}{409} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C38} 15 | \pmrelated{Path} 16 | \pmrelated{Cycle} 17 | \pmrelated{Graph} 18 | \pmrelated{PathConnected} 19 | 20 | \endmetadata 21 | 22 | % this is the default PlanetMath preamble. as your knowledge 23 | % of TeX increases, you will probably want to edit this, but 24 | % it should be fine as is for beginners. 25 | 26 | % almost certainly you want these 27 | \usepackage{amssymb} 28 | \usepackage{amsmath} 29 | \usepackage{amsfonts} 30 | 31 | % used for TeXing text within eps files 32 | %\usepackage{psfrag} 33 | % need this for including graphics (\includegraphics) 34 | %\usepackage{graphicx} 35 | % for neatly defining theorems and propositions 36 | %\usepackage{amsthm} 37 | % making logically defined graphics 38 | %%%\usepackage{xypic} 39 | 40 | % there are many more packages, add them here as you need them 41 | 42 | % define commands here 43 | \begin{document} 44 | A \emph{simple path} in a graph is a path $P=v_0 e_0 v_1\dots e_{n-1} v_n$ 45 | such that no vertex occurs twice in $P$. 46 | Some authors relax this condition by permitting $v_0=v_n$. In this case the path is usually called a cycle. 47 | %A \emph{simple path} in a graph is a path $v_0 e_0 v_1 e_1\dots e_{n-1} v_n$ 48 | %such that no vertex is repeated, except that possibly $v_0=v_1$. In the latter %case, the simple path is called a cycle. 49 | %A \emph{simple path} in a graph is a path that contains no vertex more than once. 50 | %By definition, cycles are particular instances of simple paths. 51 | %%%%% 52 | %%%%% 53 | \end{document} 54 | -------------------------------------------------------------------------------- /05C45-FleurysAlgorithm.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{FleurysAlgorithm} 4 | \pmcreated{2013-03-22 13:35:15} 5 | \pmmodified{2013-03-22 13:35:15} 6 | \pmowner{mathcam}{2727} 7 | \pmmodifier{mathcam}{2727} 8 | \pmtitle{Fleury's algorithm} 9 | \pmrecord{9}{34210} 10 | \pmprivacy{1} 11 | \pmauthor{mathcam}{2727} 12 | \pmtype{Algorithm} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | \pmrelated{EulerCircuit} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | Fleury's algorithm constructs an Euler circuit in a graph (if it's possible). \\ 42 | \begin{enumerate} 43 | \item Pick any vertex to start 44 | \item From that vertex pick an edge to traverse, considering following rule: never cross a bridge of the reduced graph unless there is no other choice 45 | \item Darken that edge, as a reminder that you can't traverse it again 46 | \item Travel that edge, coming to the next vertex 47 | \item Repeat 2-4 until all edges have been traversed, and you are back at the starting vertex 48 | \end{enumerate} 49 | By ``reduced graph'' we mean the original graph minus the darkened (already used) edges. 50 | %%%%% 51 | %%%%% 52 | \end{document} 53 | -------------------------------------------------------------------------------- /05A10-InductiveProofOfBinomialTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{InductiveProofOfBinomialTheorem} 4 | \pmcreated{2013-03-22 11:48:06} 5 | \pmmodified{2013-03-22 11:48:06} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{inductive proof of binomial theorem} 9 | \pmrecord{21}{30338} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Proof} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05A10} 15 | %\pmkeywords{number theory} 16 | 17 | \endmetadata 18 | 19 | \usepackage{amssymb} 20 | \usepackage{amsmath} 21 | \usepackage{amsfonts} 22 | \usepackage{graphicx} 23 | %%%%\usepackage{xypic} 24 | \begin{document} 25 | We prove the theorem for a ring. We do not assume a unit for the ring. 26 | We do not need commutativity of the ring, but only that $a$ and $b$ commute. 27 | 28 | When $n=1$, the result is clear. 29 | 30 | For the inductive step, assume it holds for $m$. Then for $n = m+1$, 31 | \begin{eqnarray*} 32 | (a+b)^{m+1} & = & (a+b)(a+b)^m \\ 33 | & = & (a+b)(a^m + b^m+ \sum_{k=1}^{m-1} \binom{m}{k} a^{m-k} b^k )\text{ by the inductive hypothesis} \\ 34 | & = & a^{m+1} + b^{m+1} + ab^m + ba^m + \sum_{k=1}^{m-1} \binom{m}{k} a^{m-k+1} b^k + \sum_{k=1}^{m-1} \binom{m}{k} a^{m-k} b^{k+1} \\ 35 | & = & a^{m+1} + b^{m+1} + \sum_{k=1}^m \binom{m}{k} a^{m-k+1} b^k + \sum_{k=0}^{m-1} \binom{m}{k} a^{m-k} b^{k+1} \text{ by combining terms} \\ 36 | & = & a^{m+1} + b^{m+1} + \sum_{k=1}^m \binom{m}{k} a^{m-k+1} b^k + \sum_{j=1}^m \binom{m}{j-1} a^{m+1-j} b^j \text{ let j=k+1 in second sum} \\ 37 | & = & a^{m+1} + b^{m+1} + \sum_{k=1}^m \left[ \binom{m}{k} + \binom{m}{k-1} \right] a^{m+1-k}b^k \text{ by combining the sums}\\ 38 | & = & a^{m+1} + b^{m+1} + \sum_{k=1}^m \binom{m+1}{k} a^{m+1-k}b^k \text{ from Pascal's rule} \\ 39 | \end{eqnarray*} 40 | as desired. 41 | %%%%% 42 | %%%%% 43 | %%%%% 44 | %%%%% 45 | \end{document} 46 | -------------------------------------------------------------------------------- /05C69-IndependentSetAndIndependenceNumber.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{IndependentSetAndIndependenceNumber} 4 | \pmcreated{2013-03-22 14:30:06} 5 | \pmmodified{2013-03-22 14:30:06} 6 | \pmowner{rspuzio}{6075} 7 | \pmmodifier{rspuzio}{6075} 8 | \pmtitle{independent set and independence number} 9 | \pmrecord{6}{36038} 10 | \pmprivacy{1} 11 | \pmauthor{rspuzio}{6075} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C69} 15 | \pmsynonym{stable set}{IndependentSetAndIndependenceNumber} 16 | \pmsynonym{anticlique}{IndependentSetAndIndependenceNumber} 17 | \pmrelated{Clique2} 18 | \pmdefines{independent set} 19 | \pmdefines{independence number} 20 | 21 | \endmetadata 22 | 23 | % this is the default PlanetMath preamble. as your knowledge 24 | % of TeX increases, you will probably want to edit this, but 25 | % it should be fine as is for beginners. 26 | 27 | % almost certainly you want these 28 | \usepackage{amssymb} 29 | \usepackage{amsmath} 30 | \usepackage{amsfonts} 31 | 32 | % used for TeXing text within eps files 33 | %\usepackage{psfrag} 34 | % need this for including graphics (\includegraphics) 35 | %\usepackage{graphicx} 36 | % for neatly defining theorems and propositions 37 | %\usepackage{amsthm} 38 | % making logically defined graphics 39 | %%%\usepackage{xypic} 40 | 41 | % there are many more packages, add them here as you need them 42 | 43 | % define commands here 44 | \begin{document} 45 | A set of vertices in a graph $G$ is called an {\em independent set} if there are no edges between the vertices. 46 | 47 | The {\em independence number} of a graph $G$, usually denoted by $\alpha(G)$, is the size of a maximal independent set in $G$. $\alpha(G) \geq \nu$ means that there are $\nu$ vertices with no edges between them. 48 | 49 | An independent set is sometimes called a stable set or an anticlique. 50 | %%%%% 51 | %%%%% 52 | \end{document} 53 | -------------------------------------------------------------------------------- /05C70-Matching.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Matching} 4 | \pmcreated{2013-03-22 12:40:00} 5 | \pmmodified{2013-03-22 12:40:00} 6 | \pmowner{Mathprof}{13753} 7 | \pmmodifier{Mathprof}{13753} 8 | \pmtitle{matching} 9 | \pmrecord{7}{32939} 10 | \pmprivacy{1} 11 | \pmauthor{Mathprof}{13753} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C70} 15 | \pmrelated{MaximalMatchingminimalEdgeCoveringTheorem} 16 | \pmrelated{Matching} 17 | \pmrelated{EdgeCovering} 18 | \pmrelated{Saturate} 19 | \pmdefines{maximal matching} 20 | \pmdefines{perfect matching} 21 | 22 | \endmetadata 23 | 24 | % this is the default PlanetMath preamble. as your knowledge 25 | % of TeX increases, you will probably want to edit this, but 26 | % it should be fine as is for beginners. 27 | 28 | % almost certainly you want these 29 | \usepackage{amssymb} 30 | \usepackage{amsmath} 31 | \usepackage{amsfonts} 32 | 33 | % used for TeXing text within eps files 34 | %\usepackage{psfrag} 35 | % need this for including graphics (\includegraphics) 36 | %\usepackage{graphicx} 37 | % for neatly defining theorems and propositions 38 | %\usepackage{amsthm} 39 | % making logically defined graphics 40 | %%%\usepackage{xypic} 41 | 42 | % there are many more packages, add them here as you need them 43 | 44 | % define commands here 45 | \begin{document} 46 | Let $G$ be a graph. A \emph{matching} $M$ on $G$ is a subset of the edges of $G$ such that each vertex in $G$ is incident with no more than one edge in $M$. 47 | 48 | It is easy to find a matching on a graph; for example, the empty set will always be a matching. Typically, the most interesting matchings are \emph{maximal matchings}. A maximal matching on a graph $G$ is simply a matching of the largest possible size. 49 | 50 | A \emph{perfect matching} is a matching that saturates every vertex. 51 | 52 | %%%%% 53 | %%%%% 54 | \end{document} 55 | -------------------------------------------------------------------------------- /05C45-ProofOfBondyAndChvatalTheorem.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{ProofOfBondyAndChvatalTheorem} 4 | \pmcreated{2013-03-22 14:48:33} 5 | \pmmodified{2013-03-22 14:48:33} 6 | \pmowner{taxipom}{3607} 7 | \pmmodifier{taxipom}{3607} 8 | \pmtitle{proof of Bondy and Chv\'atal theorem} 9 | \pmrecord{8}{36466} 10 | \pmprivacy{1} 11 | \pmauthor{taxipom}{3607} 12 | \pmtype{Proof} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C45} 15 | 16 | \endmetadata 17 | 18 | % this is the default PlanetMath preamble. as your knowledge 19 | % of TeX increases, you will probably want to edit this, but 20 | % it should be fine as is for beginners. 21 | 22 | % almost certainly you want these 23 | \usepackage{amssymb} 24 | \usepackage{amsmath} 25 | \usepackage{amsfonts} 26 | \usepackage{amsthm} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | \begin{document} 41 | \begin{proof} 42 | The sufficiency of the condition is obvious and we shall prove the necessity by 43 | contradiction. 44 | 45 | Assume that $G+uv$ is Hamiltonian but $G$ is not. 46 | Then $G+uv$ has a Hamiltonian cycle containing the edge $uv$. Thus there exists a path $P=(x_1,\dots,x_n)$ in $G$ from $x_1=u$ to $x_n=v$ meeting all the vertices of $G$. If $x_i$ is adjacent to $x_1$ ($2\leq i\leq n$) then $x_{i-1}$ is not adjacent to $x_n$, for otherwise 47 | $(x_1,x_i,x_{i+1},\dots,x_n,x_{i-1},x_{i-2},\dots,x_1)$ is a Hamiltonian cycle of $G$. Thus $d(x_n)\leq (n-1)-d(x_1)$, that is $d(u)+d(v)\leq n-1$, a contradiction 48 | \end{proof} 49 | %%%%% 50 | %%%%% 51 | \end{document} 52 | -------------------------------------------------------------------------------- /05C38-Girth.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{Girth} 4 | \pmcreated{2013-03-22 12:45:56} 5 | \pmmodified{2013-03-22 12:45:56} 6 | \pmowner{ariels}{338} 7 | \pmmodifier{ariels}{338} 8 | \pmtitle{girth} 9 | \pmrecord{4}{33074} 10 | \pmprivacy{1} 11 | \pmauthor{ariels}{338} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C38} 15 | \pmrelated{ChromaticNumberAndGirth} 16 | 17 | \endmetadata 18 | 19 | % this is the default PlanetMath preamble. as your knowledge 20 | % of TeX increases, you will probably want to edit this, but 21 | % it should be fine as is for beginners. 22 | 23 | % almost certainly you want these 24 | \usepackage{amssymb} 25 | \usepackage{amsmath} 26 | \usepackage{amsfonts} 27 | 28 | % used for TeXing text within eps files 29 | %\usepackage{psfrag} 30 | % need this for including graphics (\includegraphics) 31 | %\usepackage{graphicx} 32 | % for neatly defining theorems and propositions 33 | %\usepackage{amsthm} 34 | % making logically defined graphics 35 | %%%\usepackage{xypic} 36 | 37 | % there are many more packages, add them here as you need them 38 | 39 | % define commands here 40 | 41 | \newcommand{\Prob}[2]{\mathbb{P}_{#1}\left\{#2\right\}} 42 | \newcommand{\Expect}{\mathbb{E}} 43 | \newcommand{\norm}[1]{\left\|#1\right\|} 44 | 45 | % Some sets 46 | \newcommand{\Nats}{\mathbb{N}} 47 | \newcommand{\Ints}{\mathbb{Z}} 48 | \newcommand{\Reals}{\mathbb{R}} 49 | \newcommand{\Complex}{\mathbb{C}} 50 | 51 | 52 | 53 | %%%%%% END OF SAVED PREAMBLE %%%%%% 54 | \begin{document} 55 | The \emph{girth} of a graph $G$ is the length of the shortest cycle in $G$.\footnote{There is no widespread agreement on the girth of a forest, which has no cycles. It is also extremely unimportant.} 56 | 57 | For instance, the girth of any grid $\Ints^d$ (where $d>2$) is 4, and the girth of the vertex graph of the dodecahedron is 5. 58 | %%%%% 59 | %%%%% 60 | \end{document} 61 | -------------------------------------------------------------------------------- /05C90-HasseDiagram.tex: -------------------------------------------------------------------------------- 1 | \documentclass[12pt]{article} 2 | \usepackage{pmmeta} 3 | \pmcanonicalname{HasseDiagram} 4 | \pmcreated{2013-03-22 12:15:23} 5 | \pmmodified{2013-03-22 12:15:23} 6 | \pmowner{bbukh}{348} 7 | \pmmodifier{bbukh}{348} 8 | \pmtitle{Hasse diagram} 9 | \pmrecord{18}{31639} 10 | \pmprivacy{1} 11 | \pmauthor{bbukh}{348} 12 | \pmtype{Definition} 13 | \pmcomment{trigger rebuild} 14 | \pmclassification{msc}{05C90} 15 | \pmrelated{Poset} 16 | \pmrelated{PartialOrder} 17 | 18 | \endmetadata 19 | 20 | \usepackage{amssymb} 21 | \usepackage{amsmath} 22 | \usepackage{amsfonts} 23 | %%%\usepackage{xypic} 24 | \begin{document} 25 | If $(A,\leq)$ is a finite poset, then it can be represented by a \emph{Hasse diagram}, which is a graph whose vertices are elements of $A$ and the edges correspond to the covering relation. More precisely an edge from $x\in A$ to $y\in A$ is present if 26 | \begin{itemize} 27 | \item $x < y$. 28 | \item There is no $z \in A$ such that $x < z$ and $z < y$. (There are no in-between elements.) 29 | \end{itemize} 30 | If $x