Graphs and Hypergraphs
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1 Graphs and Hypergraphs CLAUDE BERGE University of Paris Translated by Edward Minieka NORTH-HOLLAND PUBLISHING COMPANY-AMSTERDAM LONDON AMERICAN ELSEVIER PUBLISHING COMPANY, INC. - NEW YORK
2 CHAPTER 1. BASIC CONCEPTS PART ONE GRAPHS 1. Graphs 3 2. Basic definitions 5 3. List of symbols 9 CHAPTER 2. CYCLOMATIC NUMBER 1. Cycles and cocycles Cycles in a planar graph 17 CHAPTER 3. TREES AND ARBORESCENCES 1. Trees and cotrees Strongly connected graphs and graphs without circuits Arborescences Injective, functional and semi-functional graphs Counting trees 42 CHAPTER 4. PATHS, CENTRES AND DIAMETERS 1. The path problem The shortest path problem Centres and radii of a quasi-strongly connected graph Diameter of a strongly connected graph Counting paths 74 CHAPTER 5. FLOW PROBLEMS 1. The maximum flow problem The compatible flow problem An algebraic study of flows and tensions The maximum tension problem 95 CHAPTER 6. DEGREES AND DEMI-DEGREES 1. Existence of a p-graph with given demi-degrees Existence of ар-graph without loops with given demi-degrees, Existence of a simple graph with given degrees 115 xi *
3 XÜ TABLE OF. CONTENTS CHAPTER 7. MATCHINGS 1. The maximum matching problem The minimum covering problem Matchings in bipartite graphs An extension of the König theorem Counting perfect matchings 142 CHAPTER 8. c-matchings 1. The maximum c-matching problem Transfers Maximum cardinality of a c-matching 155 CHAPTER 9. CONNECTIVITY 1. A-Connected graphj Articulation vertices and blocks fc-edge-connected graphs 181 CHAPTER 10. HAMILTONIAN CYCLES 1. Hamiltonian paths and circuits Hamiltonian paths in complete graphs Existence theorems for hamiltonian circuits Existence theorems for hamiltonian cycles Hamilton-connected graphs Hamiltonian cycles in planar graphs (abstract) 223 CHAPTER 11. COVERING EDGES WITH CHAINS 1. Eulerian cycles Covering edges with disjoint chains Counting eulerian circuits 239 CHAPTER 12. CHROMATIC INDEX 1. Edge colourings The Vizing theorem and related results Edge colourings of planar graphs (abstract) 267
4 ХШ CHAPTER 13. STABILITY NUMBER 1. Maximum stable sets The Turän theorem and related results a-critical graphs Critical vertices and critical edges Stability number and vertex coverings by paths 298 CHAPTER 14. KERNELS AND GRUNDY FUNCTIONS 1. Absorption number Kernels Grundy functions Nim games 318 CHAPTER 15. CHROMATIC NUMBER 1. Vertex colourings y-critical graphs The Hajos theorem Chromatic polynomials Vertex colourings of planar graphs (abstract) 355 CHAPTER 16. PERFECT GRAPHS 1. Perfect graphs Comparability graphs Triangulated graphs /-Triangulated graphs Interval graphs Cartesian product and Cartesian sum of simple graphs PART TWO HYPERGRAPHS CHAPTER 17. HYPERGRAPHS AND THEIR DUALS 1. Hypergraphs Cycles in a hypergraph Conformal hypergraphs Representative graph of a hypergraph 400 CHAPTER 18. TRANSVERSALS 1. Matchings and c-matchings 2. Transversal number I
5 Xiv CHAPTER 19. CHROMATIC NUMBER OF A HYPERGRAPH 1. Stability number and chromatic number of a hypergraph Cliques of a hypergraph Good colourings of the edges of a graph Generalizations of the chromatic number of a graph 443 CHAPTER 20. BALANCED HYPERGRAPHS AND UNIMODULAR HYPERGRAPHS 1. Strong chromatic number Balanced hypergraphs Unimodular hypergraphs Stochastic functions 469 CHAPTER 21. MATROIDS 1. Matroid on a set The Rado theorem and related results Image of a matroid Minimum weight basis 493 REFERENCES 498 INDEX OF DEFINITIONS 523 \
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