Graphs and Homomorphisms Pavol Hell Simon Fraser University, Burnaby, B.C., Canada and Jaroslav Nesetfil Charles University, Prague, The Czech Republic OXJORD UNIVERSITY PRESS CONTENTS 1 2 3 6 10 16 18 20 27 33 34 Products and retracts The product Dimension The Lovasz vector and the Reconstruction Conjecture Exponential digraphs Shift graphs The Product Conjecture and graph multiplicativity Projective digraphs and polymorphisms The retract Isometric trees and cycles Reflexive absolute retracts Reflexive dismantlable graphs Median graphs Remarks Exercises 37 37 40 43 46 47 50 57 58 60 64 68 72 76 78 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11 2.12 2.13 2.14 3 1 1 Introduction 1.1 Graphs, digraphs, and homomorphisms 1.2 Homomorphisms preserve adjacency 1.3 Homomorphisms generalize colourings 1.4 The existence of homomorphisms 1.5 Homomorphisms generalize isomorphisms 1.6 Homomorphic equivalence 1.7 The composition of homomorphisms 1.8 Homomorphisms model assignments and schedules 1.9 Remarks 1.10 Exercises The partial order of graphs and homomorphisms The partial orders C and Cs Representing ordered sets Incomparable graphs and maximal antichains in Cs Sparse graphs with specified homomorphisms Incomparable graphs with additional properties Incomparable graphs on n vertices Density Duality and gaps Maximal antichains in C 3.10 Bounds 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 XI 81 81 82 85 89 93 94 96 98 101 102 CONTENTS 3.11 Remarks 3.12 Exercises The 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 structure of composition Introduction Rigid digraphs An excursion to infinity The replacement operation Categories Representation A combinatorial obstacle to representation Some categories are not rich enough Remarks Exercises 106 107 109 109 109 115 117 122 128 132 135 138 139 5 Testing for the existence of homomorphisms 5.1 The .ff-colouring problem 5.2 Dichotomy for graphs 5.3 Digraph homomorphisms and CSPs 5.4 Duality and consistency 5.5 Pair consistency and majority functions 5.6 List homomorphisms and retractions 5.7 Trigraph homomorphisms 5.8 Generalized split graphs 5.9 Remarks 5.10 Exercises 142 142 143 151 161 166 170 178 183 186 187 6 Colouring—variations on a theme 6.1 Circular colourings 6.2 Fractional colourings 6.3 T-colourings 6.4 Oriented and acyclic colourings 6.5 Remarks 6.6 Exercises 192 192 200 210 212 218 219 References 222 Index 239