Graph Coloring Problems

Transcription

1 Graph Coloring Problems TOMMY R. JENSEN BJARNE TOFT Odense University A Wiley-Interscience Publication JOHN WILEY & SONS New York Chichester Brisbane Toronto Singapore

2 Contents Preface xv 1 Introduction to Graph Coloring Basic Definitions Graphs on Surfaces Vertex Degrees and Colorings Criticality and Complexity Sparse Graphs and Random Graphs Perfect Graphs Edge-Coloring Orientations and Integer Flows List Coloring Generalized Graph Coloring Final Remarks 23 Bibliography * 23 2 Planar Graphs Four-Color Theorem Cartesian Sequences Intersection Graphs of Planar Segments Ringel's Earth-Moon Problem Ore and Plummer's Cyclic Chromatic Number Vertex Partitionings w.r.t. Coloring Number Vertex Partitionings w.r.t. Maximum Degree The Three-Color Problem Steinberg's Three-Color Problem Grunbaum and Havel's Three-Color Problem Grbtzsch and Sachs'Three-Color Problem Barnette's Conjecture List-Coloring Planar Graphs ^ 46

3 viii Contents 2.14 Kronk and Mitchem's Entire Chromatic Number Nine-Color Conjecture Uniquely Colorable Graphs Density of 4-Critical Planar Graphs Square of Planar Graphs 50 Bibliography 51 3 Graphs on Higher Surfaces Heawood's Empire Problem Griinbaum's 3-Edge-Color Conjecture Albertson's Four-Color Problem Improper Colorings Number of 6-Critical Graphs on a Surface Toroidal Polyhedra Polynomial Coloring of Embedded Graphs Sparse Embedded Graphs Ringel's 1-Chromatic Number Borodin's Conjecture on Diagonal Coloring Acyclic Colorings Cochromatic Numbers Graphs on Pseudo-Surfaces 70 Bibliography 73 4 Degrees The Coloring Number Coloring of Decomposable Graphs Color-Bound Families of Graphs Edge-Disjoint Placements Powers of Hamilton Cycles Brooks' Theorem for Triangle-Free Graphs Graphs Without Large Complete Subgraphs ^-Chromatic Graphs of Maximum Degree k Total Coloring Equitable Coloring Acyclic Coloring Melnikov's Valency-Variety Problem Induced-Odd Degree Subgraphs Strong Chromatic Number 91 Bibliography 92 5 Critical Graphs Critical Graphs With Many Edges Minimum Degree of 4- and 5-Critical Graphs > 98

4 Contents ix 5.3 Critical Graphs With Few Edges Four-Critical Amenable Graphs Four-Critical Degree 5 Problem Large Critical Subgraphs of Critical Graphs Critical Subgraph Covering a 2-Path Noninduced Critical Subgraphs Number of Critical Subgraphs Subgraphs of Critical Graphs Minimal Circumference of Critical Graphs The Erdos-Lovasz Tihany Problem Partial Joins of Critical Graphs Vertex-Critical Graphs Without Critical Edges 105 Bibliography The Conjectures of Hadwiger and Hajos Hadwiger's Conjecture Hajos'Conjecture The (m,ri)-and [m, n]-conjectures Hadwiger Degree of a Graph Graphs Without Odd-K Scheme Conjecture Chromatic 4-Schemes Odd Subdivisions of K Nonseparating Odd Cycles in 4-Critical Graphs Minimal Edge Cuts in Contraction-Critical Graphs Kostochka 's Conjecture on Hadwiger Number 117 Bibliography Sparse Graphs Blanche Descartes'Triangle-Free Graphs Grunbaum's Girth Problem Smallest Triangle-Free/t-Chromatic Graphs Large Bipartite Subgraphs of Triangle-Free Graphs Sparse Subgraphs Number of Odd Cycle Lengths Maximum Girth of ^-Chromatic Graphs Maximum Ratio x/o> Chromatic Number of Sparse Random Graphs 128 Bibliography Perfect Graphs Strong Perfect Graph Conjecture Markosyan's Perfect Graph Problems 132

5 x Contents 8.3 Bold Conjecture Raspail (Short-Chorded) Graphs "Semistrong" Perfect Graph Conjecture Hoang's Conjecture on 2-Coloring Edges Neighborhood Perfect Graphs Monsters Square.Free Berge Graphs Weakened Strong Perfect Graph Conjecture Gyarfas' Forbidden Subgraph Conjecture Quasiperfect Graphs Perfect Graph Recognition f-perfect Graphs 144 Bibliograpfiy Geometric and Combinatorial Graphs Hadwiger-Nelson Problem Ringel's Circle Problem Sachs'Unit-Sphere Problem Sphere Colorings Graphs of Large Distances Prime Distance Graphs Cube-Like Graphs Odd Graph Conjecture Chord Intersection Graphs Gyarfas and Lehel's Triangle-Free L-Graphs ErdSs-Faber-Lovasz Problem Alon-Saks-Seymour Problem General Kneser Graphs Question of Gallai Related to Sperner's Lemma 162 Bibliography Algorithms Polynomial Graph Coloring Polynomial Approximation Even Chromatic Graphs Grundy Number Achromatic Number of a Tree On-Line Coloring Edge-Coloring Multigraphs Complexity of Directed-Graph Coloring Precedence Constrained 3-Processor Scheduling 176 Bibliography 177

6 Contents xi 11 Constructions Direct Product Wreath Product A Very Strong Product Gallai's Problem on Dirac's Construction Hajos Versus Ore Length of Hajos Proofs Hajos Constructions of Critical Graphs Construction of Hajos Generalized by Dirac Four-Chromaticity in Terms of 3-Colorability 186 Bibliography Edge Colorings Goldberg's Conjecture Jakobsen's Conjecture Seymour's r-multigraph Conjecture Weak Critical Graph Conjecture Critical Multigraph Conjecture Vizing's 2-Factor Conjecture Vizing's Planar Graph Conjecture Minimal Number of Edges in ^'-Critical Graphs Independent Sets in *'-Critical Graphs Hilton's Overfull Subgraph Conjecture The A-Subgraph Conjecture Regular Graphs of High Degree Berge and Fulkerson's Conjecture Petersen Coloring Tutte's Conjecture on 3-Edge Colorings Grb'tzsch and Seymour's Conjecture Cycle-Decomposable 4-Regular Plane Multigraphs Seymour's Planar 4-Multigraph Conjecture Uniquely 3-Edge-Colorable Planar Graphs List-Edge-Chromatic Numbers Strong Chromatic Index Vizing's Interchange Problem Scheduling Without Waiting Periods 203 Bibliography Orientations and Flows Tutte's 5-Flow Conjecture Tutte's 4-Flow Conjecture Tutte's 3-Flow Conjecture Bouchet's 6-Flow Conjecture 213

7 xii Contents Bibliography Jaeger's Circular Flow Conjecture Berge's Strong Path Partition Conjecture Berge's Directed Path-Conjecture Minimal Orientations of Critical Graphs Alon-Tarsi Orientations and Chromatic Number 14 Chromatic Polynomials Bibliography 15 Hypergraphs Bibliography Coefficients of Chromatic Polynomials Characterization of Chromatic Polynomials Chromatic Uniqueness Chromatic Equivalence Zeros of Chromatic Polynomials Beraha Conjecture Chessboard Problem Coefficients for Hypergraphs Erdos' Property B Property B(s) Finite Projective Planes Steiner Triple Systems Steiner Quadruple Systems._ Minimum-Weight 3-Chromatic Hypergraphs Positional Games Tic-Tac-Toe Square Hypergraphs Size of 3-Chromatic Uniform Cliques Monochromatic Sum-Sets Arithmetic Progressions Unprovability The Direct Product of Hypergraphs Maximal Complete Subgraphs in Perfect Graphs Coloring Triangulable Manifolds Berge's Conjecture on Edge-Coloring 16 Infinite Chromatic Graphs Sparse Subgraphs of High Chromatic Number Infinite Chromatic Subgraphs Almost Bipartite Subgraphs Laree Finite «-Chromatic Suberaohs

8 Contents xiii 16.5 Trees in Triangle-Free Graphs Unavoidable Classes of Finite Subgraphs Chromatic Subgraphs Avoiding 5-Cycles and Large Bipartite Subgraphs Connectivity of Subgraphs Set of Odd Cycle Lengths Unavoidable Cycle Lengths Coloring Number Direct Product Partition Problem of Galvin and Hajnal Small Subgraphs of Large Chromatic Number 258 Bibliography Miscellaneous Problems List-Coloring Bipartite Graphs List-Coloring the Union of Graphs " Cochromatic Number Star Chromatic Number Harmonious Chromatic Number Achromatic Number Subchromatic Number Multiplicative Graphs Reducible Graph Properties "-Colorings Game Chromatic Number Harary and Tuza's Coloring Games Coloring Extension Game Winning Hex 271 Bibliography 272 Author Index 277 Subject Index 287