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
1 - Graphs
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
A graph G consists of a set V (or V(G)) of vertices, a set E (or E(G)) of edges, and a mapping associating to each edge e ∈ E(G) an unordered pair x, y of vertices called the endpoints (or simply the ends) of e. We say an edge is incident with its ends, and that it joins its ends. We allow x = y, in which case the edge is called a loop. A vertex is isolated when it is incident with no edges.
It is common to represent a graph by a drawing where we represent each vertex by a point in the plane, and represent edges by line segments or arcs joining some of the pairs of points. One can think e.g. of a network of roads between cities. A graph is called planar if it can be drawn in the plane such that no two edges (that is, the line segments or arcs representing the edges) cross. The topic of planarity will be dealt with in Chapter 33; we wish to deal with graphs more purely combinatorially for the present.
Thus a graph is described by a table such as the one in Fig. 1.1 that lists the ends of each edge. Here the graph we are describing has vertex set V = {x, y, z, w} and edge set E = {a, b, c, d, e, f, g}; a drawing of this graph may be found as Fig. 1.2(iv).
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- Information
- A Course in Combinatorics , pp. 1 - 11Publisher: Cambridge University PressPrint publication year: 2001
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