1-Factorization Conjecture 106 2-Core of Graph 221 2-Join of Graph

Index

Cambridge University Press 978-1-107-03350-4 - Topics in Chromatic Graph Theory Edited by Lowell W. Beineke, Robin J. Wilson and Bjarne Toft Index More information

Index

1-factorization conjecture 106 anticonnected graph 139 2-core of graph 221 antihole 138 2-join of graph 143 antipath 139 2-section graph of hypergraph 237 anti-Sterboul cycle 235 3-flow conjecture 183, 185, 328, 349 arc 7 4-flow conjecture 183, 185, 348 Artemis graph 154 4-list-colourability 123 asymmetric games 315 5-flow conjecture 183, 187, 348 asymptotically almost surely 200 5-list colourability 122 average degree conjecture 107 5-list-colourability of planar graphs 327 avoidance game 321 6-realizable set 70 away game 268

a.a.s. 200 balanced edge 348 achievement game 321 balanced edge-colouring 270 activation strategy 308 balanced hypergraph 235 active vertex in game 309 balanced partition 144, 213 acyclic chromatic number 29, 306 bandwidth colouring 265 acyclic colouring 29 Barnette’s conjecture 336 acyclic list-chromatic number 29 basic graph 141, 142 acyclic orientation game 322 batch colouring problem 258 acyclically choosable graph 29 Berge graph 138 adaptive chromatic number 332 Berge trigraph 152, 153 adjacency lemma 106 Berge-Fulkerson conjecture 345 adjacent edges 2 bicolourable hypergraph 231 adjacent vertices 2 bi-hypergraph 243 adjacent vertices in hypergraph 231 bimodality 28 Albertson’s 4-colour problem 338 bipartite graph 3 algorithm 79, 81, 109, 277ff bipartite representation of hypergraph almost chromatic number 169 236 alternating path 97 bipartition algorithm 281 Andersen’s conjecture 107 block 8, 37 anti-concentration 208 block of Steiner system 246 363

Cambridge University Press 978-1-107-03350-4 - Topics in Chromatic Graph Theory Edited by Lowell W. Beineke, Robin J. Wilson and Bjarne Toft Index More information

364 Index

blossom algorithm 109 class–teacher timetabling 256, 266, 271 boundary 191 claw-free graph 60, 85, 116 bounded orientation 188 clean hole 151 Brams’s problem 337 cleaning technique 152 breadth-first ordering 282 clique 2 bridge 9 clique number 2, 45 Brooks R. L. 50 clique partition graph 331 Brooks’s theorem 10, 36ff, 37, 39 closed walk 3 broom 331 co-chromatic number 261 brute force algorithm 277 co-colouring 261 collapsible graph 191, 193 cardinality of edge 231 colour bounded hypergraph 248 Cartesian product 5, 311 colour class of hypergraph 231 centre 143 colour list 12 characteristic polynomial 65 colourable mixed hypergraph 243 choice number 12, 115 colour-critical 19 choice ratio 130 colouring games 304ff choice-perfect graph 133 colouring graphs on surfaces 13ff choosability 88 colouring number of graph 224, 307 choosability conjectures 118 colouring number of hypergraph 231 choosable graph 12, 115 colouring of graph 9, 139 chromatic index 11, 94, 116, 223 colouring of hypergraph 231 chromatic index of hypergraph 235 colouring of mixed hypergraph 233 chromatic inversion of hypergraph 244 colouring random graphs 199ff chromatic number of graph 9, 131 colouring rate 208 chromatic number of hypergraph 231 colouring type 167 chromatic number of surface 11ff colouring with preferences 262 chromatic number of the plane 162, 165 colouring-flow duality 25 chromatic polynomial 56ff combinatorial algorithm 156 chromatic polynomial of hypergraph 231 combinatorial Nullstellensatz 119 chromatic root 60 comet 331 chromatic scheduling 255ff compactness 273 chromatic spectrum of hypergraph 244 complement 2-join 143 chromatic-choosable graph 319 complement of graph 1 circuit 3 complete bipartite graph 4 circular chromatic number 26 complete graph 4 circular colouring 26 complete subset 139 circular flow 189 complete uniform hypergraph 231 circular flow index 26 component 3 circular hypergraph 231, 240 concentration 205 class 1 graph 105, 223 conjugate 63 class 2 graph 105, 223 connected component 3 classification problem 96, 105 connected graph 3

Cambridge University Press 978-1-107-03350-4 - Topics in Chromatic Graph Theory Edited by Lowell W. Beineke, Robin J. Wilson and Bjarne Toft Index More information

Index 365

connectivity 9 deletion of vertex 5 consecutive colouring 274 deletion/contraction lemma 57 contains 138 dense random graphs 202 contractible 6 density of graph 344 contractible configuration 191 depth-first ordering 282 contraction algorithm 284 diameter of graph 3 contraction of an edge 6,155 dichotomy of edge-colouring 108 contraction-critical graph 75 digraph 7, 118 core of graph 106, 221 Dirac’s theorem 10 cover graph 323 directed graph 7 covering number of hypergraph 332 disconnected graph 3 C-perfect hypergraph 239 disjunctive pair 261 critical edge 106 distance 3 critical graph 10, 40, 41, 107, 342 domain of colouring 96 critical hypergraph 235 dominant pair 155 critical multigraph conjecture 107 double diamond 147 crown graph 280 double split graph 141 cubic graph 2 double star cutset 144 cut-edge 9 double-critical graph 331 cut-vertex 8 doubled graph 142 cycle 3, 58 dynamic programming 285 cycle double cover 194 cycle graph 4 edge 1 cycle in hypergraph 235 edge of hypergraph 230 cycle matroid 65 edge-chromatic number 11 cycle space minor 196 edge-colouring 11, 94ff, 116, 223, 266, cyclic compactness 274 344 cyclic interval colouring problem 258 edge-colouring algorithm 298 cycloid 240 edge-colouring games 312 edge-colouring property in hypergraph Dawson’s chess 322 236 decomposition 138, 142 edge-connected 186 decomposition theorem 138 edge-connectivity 9 defective colour 103 edge-critical graph 106 deficiency of graph 274 edge-disjoint placements 339 degeneracy 281 edge-list colouring 125 degenerate chromatic number 31 edge-set 1 degenerate colouring 30, 31 edge-width 21 degenerate graph 117 Edmonds’s blossom algorithm 109 degenerate list-chromatic number 31 Edmonds’s matching polytope theorem degree of vertex 2 108 degree of vertex in hypergraph 236 elementary set 96 deletion of edge 5 end-block 38

Cambridge University Press 978-1-107-03350-4 - Topics in Chromatic Graph Theory Edited by Lowell W. Beineke, Robin J. Wilson and Bjarne Toft Index More information

366 Index

end-vertex 2 fractional paint number 320 equitable chromatic number 51, 222 fringe of graph 221 equitable chromatic threshold 222 equitable colouring 51 Gallai–Roy theorem 260 equivalence of flows 183 game Grundy index 313 Erdos–Faber–Lov˝ asz´ problem 330 game-chromatic number 305, 315, 337 Erdos–Lov˝ asz´ Tihany conjecture 331 game-colouring number 307, 315 Erdos–R˝ enyi´ random graphs 201 game-perfect graphs 311 Euler genus 18 games on digraphs 314 Euler’s formula 18 -connected 191 Eulerian digraph 120 generalized theta graph 62 Eulerian graph 3 geometric graph 161ff Eulerian trail 3 girth of graph 3 even 2-join 143 Gn,m model 200 even pair 154 Gn,p model 199 even subgraph 348 Goldberg’s conjectures 341, 345 even-faced embedding 27 Goldberg–Seymour conjecture 95, 101, exhaustive search algorithm 277 108 expose-and-merge algorithm 208 good colouring of hypergraph 237 extra-k-connected graph 76 good partition 142 extreme graph 103 graph 1 graph colouring algorithms 277ff factorization 106 graph minor 74 fan 98, 290 greedy algorithm 204 fan inequality 100 greedy colouring 279 Fan–Raspaud conjecture 346 greedy colouring games 312 feasible partition 244 Grotzsch’s¨ theorem 23 finite projective plane 246 group flow 181 finitization of the four-colour problem group structure 193 334 growth constant 219 five-colour theorem 183 Grundy index 313 flow 181ff, 348 Grundy number 304 flow double covering 195 Gyarf´ as’s´ forbidden subgraph conjecture flow index 26, 189 330 flow polynomial 64 Gyarf´ as–Sumner´ conjecture 331 flow problems 348 forbidden cycle 123 Hadwiger’s conjecture 73ff, 221, 328 forest 3 Hajos´ sum 41 four-colour problem 13, 333 Hamiltonian graph 3 four-colour theorem 16, 183, 327, 328 Hasse diagram 250 fractional choice number 320 Heawood problem 14 fractional chromatic index 108 Heawood’s formula 18 fractional colouring 89 Helly property 332

Cambridge University Press 978-1-107-03350-4 - Topics in Chromatic Graph Theory Edited by Lowell W. Beineke, Robin J. Wilson and Bjarne Toft Index More information

Index 367

hereditary property 224 join 2 Hilton graph 106 join of graphs 5 Hilton’s overfull conjecture 106 historical basic classes 141 k-choosable graph 12, 14, 115 hole 138 k-colourable graph 9 home game 268 k-colour-critical 19 homeomorphic graphs 6 k-connected graph 8 homogeneous pair 143 k-contraction-critical graph 75 host graph of hypergraph 231 k-critical graph 10, 19, 40, 342 hyperedge 230 k-degenerate colouring 30 hypergraph 230 k-degenerate graph 117 hypergraph colouring 230ff k-dimensional tree 4 hyperstar 231 k-edge-colourable graph 11 hypertree 231, 240 Kempe change 38, 97 Kempe equivalent colourings 110 kernel 118 immersion 87 kernel method 317 imperfect graph 137 kernel-perfect digraph 118 imperfect hypergraph 239 k-flow 182, 348 impoverished graph 155 Kierstead path 98 improper chromatic number 225 Klein bottle 20 improper colouring 225, 242 k-linked graph 77 inaccessible cardinal 176 k-list-colourable 12 incident 231 k-list-critical graph 52 incident vertex and edge 2 Konig¨ property 236 inclusion–exclusion algorithm 289 Konig’s¨ theorem 12, 94 in-degree 7, 118 k-regular graph 2 independence number 2, 231 k-tree 4 independence number conjecture 107 largest-first ordering 280 independent set 2, 85 Lawler’s algorithm 287 independent set in hypergraph 231 L-colourable 115 independently-1-tough graph 62 L-colouring 12, 14, 114, 131 induced subdigraph 118 L-critical 132 induced subgraph 5 leaf 2 induced subhypergraph 239 leap for a path 148 inflation 131 l-edge-connected graph 9 integer flow 181ff length of walk 3 internally disjoint paths 8 line graph 116 interval hypergraph 231, 240 linear intersection number 330 I-regular colouring of hypergraph 238 list 14 isolated vertex 2 list-assignment 114 isomorphic graphs 2 list-chromatic index 116, 347

Cambridge University Press 978-1-107-03350-4 - Topics in Chromatic Graph Theory Edited by Lowell W. Beineke, Robin J. Wilson and Bjarne Toft Index More information

368 Index

list-chromatic number 12, 14, 50, 115, multiple edges of hypergraph 230 131 multiple of graph 103 list-colourable 115 multivariate Tutte polynomial 66 list-colouring 12, 14, 88, 114ff list-colouring of hypergraph 238 near-triangulation 15 list-colouring problem 258 neighbour 2 list-critical graph 52, 132 neighbourhood 2 list-edge-colouring conjecture 116 neighbourhood in hypergraph 231 list-square-colouring conjecture 117 neighbours in hypergraph 231 local tension 27 Node Kayles 322 locally planar graph 21 non-orientable surface 18 log concavity 59 non-path 2-join 143 loop 1 non-restrictive function 248 Lovasz’s´ perfect graph theorem 141 non-separable graph 8 lower chromatic number of hypergraph nowhere-zero ﬂow 182, 184 244 null graph 4

major vertex 149 odd 2-join 143 maker–breaker game 305 odd complete minor 82 marked vertex in game 308 odd conjecture 82 matching polytope 108 odd edge-connected 186 matroid 65 odd wheel 155 maximal independent set 286 oddness of graph 187 maximal Kierstead 98 Ohba’s conjecture 319 maximum average degree 122 online chromatic number 304 maximum density 217 open shop scheduling 266 maximum independent set 38 optimal ordering 280 measurable chromatic number 172 order of graph 1 Menger’s theorem 8, 9 order of hypergraph 230 Metropolis algorithm 291 order of separation 76 minimal list-critical 132 Ore’s conjecture 43 minimally imperfect graph 138 orientable surface 18 minor 6, 74, 328 orientation 118 minor vertex 149 oriented game-chromatic number 314 missing colours 96 out-degree 7, 118 mixed graph colouring 260 overfull graph 106 mixed hypergraph 233, 243 mod-k-ﬂow 182 packing density 218 modulo k-orientation 186, 188, 190 paintability 316 monostar 239 painting game 317 monotone graph invariant 40 palette restriction 283 Mosers spindle 163 pancyclic graph 3 multicolouring problem 256 parallel reduction operation 67 multiple edges 1 parity-breaking path 83

Cambridge University Press 978-1-107-03350-4 - Topics in Chromatic Graph Theory Edited by Lowell W. Beineke, Robin J. Wilson and Bjarne Toft Index More information

Index 369

partial edge-colouring 96 replicating a vertex 140 partial subhypergraph 236 replication lemma 140 partite sets 3 restriction of hypergraph 235 partitioning a set 164 Richter’s planar graph problem 126, 127 path 3, 138 ring graph 121 path cobipartite graph 154 Ringel’s earth–moon problem 336 path double-split graph 154 Ringel–Youngs theorem 11 path graph 4 Rounded vector colouring 294 perfect graph 137ff Roussel–Rubio lemma 148 perfect graph theorem 140 perfect graph theorem for hypergraphs satisfiability 299 236 school timetabling problem 256, 266 perfect hypergraph 239 separation 76 phantom vertices 248 sequential colouring algorithm 37 phase transition 201, 211 series reduction operation 67 planar bipartite graph 121 set of chromatic cardinalities 176 planar graph games 310 set system 230 planar graph growth constant 219 Shannon’s theorem 95 planar hypergraph 237 sharp threshold 211 planarity gadget 298 shortest-path detector 152 polychromatic number of the plane 162, skew cutset 144 163, 167 skew partition 144 polymer 63 smallest-last ordering 281 Potts model 201 spanning subgraph 5 Potts model partition function 66 sparse random graphs 208 precedence constraints 260 SPGT 137 precolouring extension 128, 321 spilling 263 probability colouring 343 split colouring problem 262 product of flows 184 split graph 262 proper class 328 sports scheduling 268 P-uniform graph 343 square of graph 117, 337 random Cayley graph 216 stability number 2, 203, 231 random geometric graph 217 stable set 2 random graph theory 199 stable set in hypergraph 231 random regular graph 214 stably bounded hypergraph 234, 247 randomized rounding 294 star chromatic number 31 realization of a trigraph 153 star colouring 31 realizing a distance 162 star cutset 143 reduction algorithms 297 Stechkin’s construction 167 Reed’s conjecture 341 Steinberg’s 3-colour problem 335 regular graph 2 Steinberg’s conjecture 125 regular infinite cardinal 176 Steiner system 246 relaxed games 316 strong choice-perfect graph 132

Cambridge University Press 978-1-107-03350-4 - Topics in Chromatic Graph Theory Edited by Lowell W. Beineke, Robin J. Wilson and Bjarne Toft Index More information

370 Index

strong chromatic number of hypergraph Tutte’s conjectures 182 237 Tutte’s flow conjectures 328, 348, 349 strong colouring of hypergraph 237 type of graph 27 strong component 7 strong digraph 7 underlying graph 118 strong limit 176 uniform hypergraph 231 strong perfect graph theorem 137, 138, uniform inflation 118, 131 328 union of graphs 5 strongly connected digraph 7 uniquely colourable hypergraph 245 subdivision of an edge 6 unit-distance graph 162 subdivisions 86 unsolved graph-colouring problems subgraph 5 327ff subgraph conditioning 215 upper chromatic number of hypergraph subgraph expansion of chromatic 238 polynomial 59 useful decomposition 142 sum of flows 184 vector chromatic number 293 support 182 vector colouring 292 surface non-separating edge-width 21 vertex colouring 9 switchable pair 153 vertex of digraph 7 vertex of graph 1 Tashkinov tree 97, 98 vertex of hypergraph 230 Tashkinov’s theorem 98 vertex-set 1 tension 27 Vizing fan 98 thickness of graph 336 Vizing’s adjacency lemma 106 three-colour theorem 183 Vizing’s algorithm 290 threshold 211 Vizing’s conjectures 107 total chromatic number 223, 341 Vizing’s theorem 95 total colouring 223 total colouring conjecture 341 walk 2 total graph 116 weak colouring of hypergraph 232 totally odd subdivision 86 weak perfect graph theorem 140, 236 tough graph 61 weighted graph 131 traceable graph 3 wheel 58 trail 3 Whitney expansion 287 transversal of hypergraph 236 width of tree decomposition 79 tree 3, 58 Wigderson’s algorithm 283 tree decomposition 79 WKBGSF 157 tree-width 4 Wonderful lemma 148 triangle-free graph 124 worst-case asymptotic analysis 278 triangularly connected 192 trigraph 153 Zermelo–Fraenkel–Solovay system of trivial graph 1 axioms 176 Tutte plane 69 zero-free interval 61 Tutte polynomial 68 zero-sum 191