Book contents
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- 1 Graphs
- 2 Trees
- 3 Colorings of graphs and Ramsey's theorem
- 4 Turán's theorem and extremal graphs
- 5 Systems of distinct representatives
- 6 Dilworth's theorem and extremal set theory
- 7 Flows in networks
- 8 De Bruijn sequences
- 9 Two (0, 1, ⋆) problems: addressing for graphs and a hash-coding scheme
- 10 The principle of inclusion and exclusion; inversion formulae
- 11 Permanents
- 12 The Van der Waerden conjecture
- 13 Elementary counting; Stirling numbers
- 14 Recursions and generating functions
- 15 Partitions
- 16 (0, 1)-Matrices
- 17 Latin squares
- 18 Hadamard matrices, Reed–Muller codes
- 19 Designs
- 20 Codes and designs
- 21 Strongly regular graphs and partial geometries
- 22 Orthogonal Latin squares
- 23 Projective and combinatorial geometries
- 24 Gaussian numbers and q-analogues
- 25 Lattices and Möbius inversion
- 26 Combinatorial designs and projective geometries
- 27 Difference sets and automorphisms
- 28 Difference sets and the group ring
- 29 Codes and symmetric designs
- 30 Association schemes
- 31 (More) algebraic techniques in graph theory
- 32 Graph connectivity
- 33 Planarity and coloring
- 34 Whitney Duality
- 35 Embeddings of graphs on surfaces
- 36 Electrical networks and squared squares
- 37 Pólya theory of counting
- 38 Baranyai's theorem
- Appendix 1 Hints and comments on problems
- Appendix 2 Formal power series
- Name Index
- Subject Index
5 - Systems of distinct representatives
Published online by Cambridge University Press: 05 June 2012
- Frontmatter
- Contents
- Preface to the first edition
- Preface to the second edition
- 1 Graphs
- 2 Trees
- 3 Colorings of graphs and Ramsey's theorem
- 4 Turán's theorem and extremal graphs
- 5 Systems of distinct representatives
- 6 Dilworth's theorem and extremal set theory
- 7 Flows in networks
- 8 De Bruijn sequences
- 9 Two (0, 1, ⋆) problems: addressing for graphs and a hash-coding scheme
- 10 The principle of inclusion and exclusion; inversion formulae
- 11 Permanents
- 12 The Van der Waerden conjecture
- 13 Elementary counting; Stirling numbers
- 14 Recursions and generating functions
- 15 Partitions
- 16 (0, 1)-Matrices
- 17 Latin squares
- 18 Hadamard matrices, Reed–Muller codes
- 19 Designs
- 20 Codes and designs
- 21 Strongly regular graphs and partial geometries
- 22 Orthogonal Latin squares
- 23 Projective and combinatorial geometries
- 24 Gaussian numbers and q-analogues
- 25 Lattices and Möbius inversion
- 26 Combinatorial designs and projective geometries
- 27 Difference sets and automorphisms
- 28 Difference sets and the group ring
- 29 Codes and symmetric designs
- 30 Association schemes
- 31 (More) algebraic techniques in graph theory
- 32 Graph connectivity
- 33 Planarity and coloring
- 34 Whitney Duality
- 35 Embeddings of graphs on surfaces
- 36 Electrical networks and squared squares
- 37 Pólya theory of counting
- 38 Baranyai's theorem
- Appendix 1 Hints and comments on problems
- Appendix 2 Formal power series
- Name Index
- Subject Index
Summary
We first give two different formulations of a theorem known as P. Hall's marriage theorem. We give a constructive proof and an enumerative one. If A is a subset of the vertices of a graph, then denote by г(A) the set ∪a∈Aг(a). Consider a bipartite graph G with vertex set X ∪ Y (every edge has one endpoint in X and one in Y). A matching in G is a subset E1 of the edge set such that no vertex is incident with more than one edge in E1. A complete matching from X to Y is a matching such that every vertex in X is incident with an edge in E1. If the vertices of X and Y are thought of as boys and girls, respectively, or vice versa, and an edge is present when the persons corresponding to its ends have amicable feelings towards one another, then a complete matching represents a possible assignment of marriage partners to the persons in X.
Theorem 5.1.A necessary and sufficient condition for there to be a complete matching from X to Y in G is that |г(A) > |A| for every A ⊆ X.
Proof: (i) It is obvious that the condition is necessary.
(ii) Assume that |г(A)| ≥ |A| for every A ⊆ X. Let |X| = n, m < n, and suppose we have a matching M with m edges. We shall show that a larger matching exists.
- Type
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- Information
- A Course in Combinatorics , pp. 43 - 52Publisher: Cambridge University PressPrint publication year: 2001