Topics in Topological Graph Theory Edited by Lowell W

Cambridge University Press 978-0-521-80230-7 - Topics in Topological Graph Theory Edited by Lowell W. Beineke and Robin J. Wilson Index More information

Index

absolute Galois group, 222 blocks of a transversal design, 269 accumulation point, 304 blocks of a triple system, 269 action of a group, 24, 226 book embedding, 327 adjacent edges, 2 boundary walk, 20 adjacent vertices, 2 bounded automorphism, 294 afﬁne plane, 253 bouquet, 5, 50 algorithms, 30, 73, 330 bow-tie operation, 72 almost-4-connected graph, 306 branch decomposition, 84 almost homocyclic group, 216 branch set, 191 almost planar graph, 144 branched covering, 26, 191 almost-transitive graph, 293 bridge, 9, 89 angle transformation, 169 Brooks’s theorem, 9 apex graph, 76, 331 Burnside’s lemma, 184 arc, 10, 200 arc-transitive graph, 2 atom, 297 cactus, 38 attach an ear, 58 cage, 252 attach serially, 58 Cartesian product, 6 automorphism, 2, 22 Cayley graph, 28, 225 automorphism group, 2 Cayley map, 29, 210, 227 automorphism of a map, 201, 305 C-disc embedding, 68 average crossing number, 143 cellular embedding, 14, 20 average genus, 56 chiral map, 206 choosable graph, 126 chromatic index, 9 balanced automorphism, 213 chromatic number of a graph, 9 balanced Cayley map, 211 chromatic number of a map, 22 balanced representation, 155 chromatic number of a surface, 16, 22 bar, 46 circle packing representation, 66 bar-amalgamation, 46 circuit, 3 barycentric representation, 158 circular embedding, 20 barycentric subdivision, 169 circular embedding conjecture, 20, 320 base graph, 27, 182 closed surface, 19 bead of a necklace, 38 closed walk, 3 biembeddable pair, 271 closed-end ladder, 48 biembedding, 271 cobblestone path, 5, 49 Bilinski diagram, 307 co-degree of a face, 304 bipartite graph, 3 colony, 326 bisection width, 138 colouring a graph, 9, 111 blockage, 77 combinatorial conﬁguration, 174 blocking curves, 77 combinatorial current graph, 26

341

342 Index

complement, 2 Desarguian plane, 250 complete bipartite graph, 5 d-fibre, 292 complete circuit, 282 diagonal curve, 122 complete graph, 4 diameter, 3 complete k-partite graph, 5 digraph, 10 component, 3 dilation coefficient, 154 concentric Bilinski diagram, 207 dipole, 5, 51 configuration embedding, 256 directed edge, 10 congestion of an edge, 138 directed graph, 10 congruent embeddings, 53 direction, 19 connected component, 3 disc, 175 connected covering, 185 disc embedding, 68 connected graph, 3 disc representation, 329 connectivity, 9 disconnected graph, 3 connectivity of an infinite graph, 296 disjoint crossing paths, 68 contractible, 8 displayed sets, 85 contraction of an edge, 8 distance, 3, 118 convex representation, 165 distribution of inverses, 211 core-free permutation group, 202 dominated end, 291 co-tree, 36 double ray, 290 covering, 26, 182 drawing of a graph, 135, 314 covering digraph, 27 duad, 262 covering graph, 27, 182 dual edge, 21 covering isomorphism, 182 dual embedding, 21 covering projection, 26, 181 dual map, 201 covering transformation, 182 dual vertex, 21 covering transformation group, 182 dual-width of an embedding, 146 cover, 327 Coxeter graph, 207 E-adjacent embeddings, 59 Cremona–Richmond geometry, 262 ear decomposition, 35 critical graph, 114 Earth–Moon colouring, 326 crosscap, 12 edge, 1 crosscap distribution polynomial, 47 edge fibre, 28 crosscap number, 16, 20, 47 edge-accumulation point, 303 crosscap range, 20, 47 edge-connectivity, 9 crossing number, 11, 23, 133, 136, 314 edge-homogeneous map, 308 crossing-critical graph, 139 edge-set, 1 crossing-free curves, 77 edge-symbol, 308 cubic graph, 2 edge-transitive graph, 2 current graph, 26, 281 edge-width, 324 cut-edge, 9 eigenpolytope, 177 cut-vertex, 8 eigenvector method, 156 cycle, 3 element of a geometry, 174 cycle double cover conjecture, 20, 320 elementary Abelian p-group, 216 cycle graph, 4 embeddable, 14 cycle rank, 34 embedding, 10, 14, 20, 153 cycle type, 309 embedding extension problem, 74 embedding a grid, 107 embedding in the plane, 10 dart group, 24, 200 embedding of a triple system, 270 decomposition of a graph, 84 embedding theory, 22 deficiency of a graph, 36 end vertex, 2 degree of a vertex, 2, 19 endpoint of a curve, 175 deletion of a vertex, 7 endpoint of an edge, 19 deletion of an edge, 8 end of an ear, 58 depth of trap, 59 energy function, 155 d-equivalence, 292 enumeration, 29 Desargues geometry, 258 equivalence, 292

Index 343

equivalent representations, 154 geometrical crossing number, 316 equivalent rotation systems, 24 geometrical graph, 328 Euler characteristic, 14, 21 geometrical thickness, 326 Euler genus, 64, 107, 226 girth of a graph, 3 Euler genus problem, 75 good drawing, 171, 314 Euler’s formula, 11, 14, 21 graph, 1, 19 Eulerian graph, 3 graph drawing, 63, 151 Eulerian trail, 3 graph minor, 31 even component, 36 Graph Minors Project, 81 evenly embedded graph, 115 graph nodal domain theorem, 157 excluded minor, 69 graph on a surface, 14 exponential growth, 300 graph representation, 153 extended edge-symbol, 310 grid, 82 Grötzsch’s theorem, 124 group of a transversal design, 269 face, 11, 14, 20, 164 growth degree, 300 face boundary walk, 20 Grünbaum colouring, 323 face-transitive map, 305 face-width, 117, 324 factorization, 213 Hajós join, 114 faithful action, 24 Halin graph, 322 Fano plane, 174, 245 Hamiltonian graph, 3 Fáry’s theorem, 152 handle, 12 fibre, 28, 292, 328 Heawood map colour problem, 23, finite projective plane, 247, 250 63, 112 finite-connectivity, 295 Heawood number, 23 flag, 25, 163, 174, 203, 264 Heffter’s first difference problem, 282 flag graph, 165 hereditary, 318 flag-simple, 164 high-end colouring, 113 flat type, 24 homeomorphic graphs, 8 flow, 323 homeomorphically irreducible tree, 322 F -map, 205 homogeneous representation, 176 fold number, 328 Hurwitz system, 191 forbidden family, 17 Hurwitz’s theorem, 232 forbidden subgraph, 69 force-directed placement, 160 forest, 4 four-colour theorem, 22, 322 immersion of a graph, 138 fragment, 297 incidence geometry, 174 framework, 177 incidence relation, 174 fullerene, 156 incidence structure, 174 fundamental polygon, 172 incident vertex and edge, 2 independence ratio, 117 independent diagonal curve, 122 general rotation system, 24 independent odd crossing number, 136 generalized Laplacian, 156 induced subgraph, 7 generalized Petersen graph, 143 infinite-connectivity, 296 generalized pseudosurface, 248 inserting an edge, 35 generalized quadrangle, 262 integer-length edge, 328 generic iterative graph representation algorithm, intermediate growth, 300 162 intersecting line segments, 329 genus, 15, 20 irredundant generating set, 236 genus distribution polynomial, 46 isolated vertex, 2 genus of a geometry, 249 isomorphic branched coverings, 191 genus of a group, 28, 225 isomorphic coverings, 182 genus problem, 75 isomorphic graphs, 2, 22 geodesic, 332 isomorphism of embeddings, 22 geodetic edge, 306 isomorphism of graphs, 2, 22

344 Index

join, 2 locally planar embedding, 113 join of graphs, 6 loop, 1 JW group, 241 low-end colouring, 116

k-choosable graph, 126 map, 163 k-chromatic graph, 9 map automorphism, 201, 305 k-colourable graph, 9 map colour problem, 317 k-colour-critical graph, 114 map colour theorem, 16 k-connected graph, 9, 100 map on a surface, 22 k-crossing-critical graph, 139 map theory, 22 k-degenerate graph, 126 Maschke’s characterization, 235 k-dimensional tree, 4 maximum crosscap number, 20, 47 k-edge-colourable graph, 9 maximum genus, 20, 34, 46 k-edge-connected graph, 9 medial map, 167, 221 k-flow, 323 Mendelsohn triple system, 260, 270 K-graph, 71 Menger graph, 248 k-grid, 82 Menger’s theorem, 9 meridian, 122 Kirchhoff’s current law, 281 M Kirkman triple system, 260 -flag simple map, 164 k-list-colourable graph, 126 minimal forbidden minor, 69 k-regular graph, 2 minimal forbidden subgraph, 69 k-separation, 69 minimum crosscap number, 20, 47 k-tree, 4 minimum genus, 20, 46 Klein bottle, 12 minor, 8, 31, 69, 82 minor ordering, 318 Kuratowski’s theorem, 11, 64 minor-closed family, 331 Kuratowski-type theorem, 38 minor-minimal forbidden family, 17 mixed type, 309 Möbius function, 185 ladder, 48 Möbius strip, 12 Laplace method, 156 monodromy group, 24, 25, 200 Laplacian, 156 morphism of maps, 201 large edge-width embedding, 331 multiple edges, 1 lattice, 301 leapfrog transformation, 169 ledge numbers, 216 natural projection, 28 left action of a group, 24 necklace, 5, 38, 56 left regular representation, 182 negative support, 157 length of walk, 3 neighbour, 2 Levi graph, 165, 174, 248 neighbourhood, 2, 181 LEW embedding, 331 neighbourhood graph, 270 LEW weight function, 331 neighbourhood system, 292 lexicographic product, 6 never-increasing sequence, 95 lifting an embedding, 29 non-degenerate representation, 153 line of an incidence structure, 174 non-orientable genus, 16, 226, 317 lineal incidence structure, 174 non-orientable genus distribution, 47 linear crossing number, 316 non-orientable genus problem, 75 linear drawing, 316 non-orientable surface, 12 linear genus, 172 non-separable graph, 8 linear growth, 300 normalized assignment, 183 linear thickness, 326 nowhere-zero flow, 323 linked decomposition, 89 number of embeddings, 46, 47 linked edges, 94 list-colourable graph, 126 local rotation, 315 obstruction, 63, 318 locally bipartite embedded odd component, 36 graph, 115 odd crossing number, 136 locally finite graph, 289 one-way infinite path, 290

Index 345

oppositely oriented map, 201 polynomial growth, 300 orbit-counting theorem, 184 positive support, 157 order of a blockage, 77 PQ-tree, 65 order of a graph, 1 presentation, 29 order of a projective plane, 250 primal embedding, 21 ordinary derived graph, 182 projective plane, 12, 176, 250 ordinary voltage assignment, 182 proper minor, 82 orientable genus, 317 proper separation, 69 orientable genus range, 20, 46 pseudosurface, 248, 319 orientable surface, 12, 19 pure, 52 orientation-preserving automorphism 24, 201 orientation-reversing automorphism, 24, 201 oriented map, 200 quadrangulation, 14, 116 oriented surface, 19 quadratic growth, 300 orthonormal representation, 155 quasi-geodetic ray, 292 outer-embeddable graph, 67 outerplanar graph, 66 overlap matrix, 52 rank of a graph, 302 ray, 290 ray type, 309 real projective plane, 176 page, 327 rectilinear crossing number, 145, 316 page number, 327 rectilinear drawing, 316 pair crossing number, 136 rectilinear embedding, 328 Pajek program, 160 reflection, 205 Pappus graph, 176 reflexible, 205 Pappus’s geometry, 262 region, 10, 14, 20 partite sets, 4 regular action, 24 Pasch configuration, 262 regular Cayley map, 212 path, 3 regular covering, 26, 182 path graph, 4 regular embedding, 264 perfect matching, 321 regular graph, 2, 19 periodic graph, 302 regular map, 205 peripheral cycle, 158 regular orientable map, 202 permutation derived graph, 182 regular representation, 182 permutation voltage assignment, 182 regular surface branched covering, 193 permutation voltage graph, 27 regular voltage assignment, 27 Petersen colouring conjecture, 321 regular voltage graph, 27 Petrie cycle, 218 relator, 29 Petrie dual, 218 representation, 153 Petrie path, 218 representation of a geometry, 175 Petrie walk, 308 representation polytope, 177 Petrie-orbits, 309 representativity, 117, 146, 324 piecewise-linear structure, 19 r-fold covering, 182 pinch point, 319 Riemann–Hurwitz equation, 229 planar cover, 327 Ringel’s theorem, 16 planar dilation coefficient, 154 Ringel–Youngs theorem, 15 planar graph, 10 Robertson–Seymour theorem, 17, 70 planar map, 303 root of Bilinski diagram, 307 planar representation, 153 rooted forest, 95 planar tile, 141 rooted minor, 96 planarity problem, 22 rotation, 23, 315 planarizing set, 118 rotation scheme, 23 plane graph, 10 rotation system, 23 plane representation, 153 rotational projection, 170 r Poincaré dual embedding, 21 R -representation, 153 point of incidence structure, 174 points of a transversal design, 269 points of a triple system, 269 Schlegel drawing, 159

346 Index

Schlegel representation, 159 tetrahemihexahedron, 166 semi-edge, 200 theory of algorithms, 63 separating cycles, 325 thick end, 291 separator, 296 thickness of a graph, 11, 326 serial attachment, 58 thin end, 291 set of planarizing cycles, 118 thrackle, 316 sheet of a spindle, 319 thrackle conjecture, 316 sign, 24 tile, 141 signature, 231 tile crossing number, 141 simple curve, 175 tile drawing, 141 simple graph, 1 T -normalized assignment, 183 simple triple system, 269 topological end, 293 sink of an ordering, 64 topological model, 248 size of a face, 11 topological ordering, 318 skeleton, 164 torso, 69 skew morphism, 217 total embedding distribution, 51 source of an ordering, 64 traceable graph, 3 S-outer-embeddable graph, 67 trail, 3 spanning subgraph, 6 transitive, 24 spatial representation, 153 translatable fibre, 302 sphere representation, 330 transversal design, 269 spindle, 319 trap, 59 spindle surface, 73 tree, 4 spine, 327 tree decomposition, 84 spring embedder, 160 tree-width, 4, 86 spring embedding algorithm, 66 triangle group, 203 stabilizer, 24, 201 triangulation, 14, 317 standard line, 251 tripod, 68 Steiner triple system, 246, 260, 269 Turán’s brickyard problem, 134, 315 Steinitz’s theorem, 11, 152 Tutte drawing, 159 st-ordering, 64 twisted type, 24 straightness of a ray, 292 two-fold triple system, 269 stratified graph, 59 two-way infinite path, 290 stratum, 59 type, 24, 174 strong component, 10 type of Cayley map, 211 strong homogeneous representation, 176 type of necklace, 6 strong nodal domain, 157 strong representation, 153 strong symmetric genus, 29, 237 underlying topological space, 19 strongly cellular embedding, 20 Unendlichkeitslemma, 291 unimodal sequence, 55 strongly connected digraph, 10 union of graphs, 6 strongly symmetric embedding, 227 unit-cost RAM, 65 strongly unimodal sequence, 55 unit-distance graph, 154, 332 subdivision of an edge, 8 universal cover, 173 subexponential growth, 300 universally concentric Bilinski diagram, 307 subgraph, 6 upper-embeddable graph, 39 support, 157 suppression of a vertex, 318 surface, 12 V -adjacent, 59 surgery, 26 valence, 19 symmetric embedding, 226 vertex, 1, 10, 164 symmetric Euler genus, 227 vertex addition algorithm, 65 symmetric map, 199 vertex fibre, 28 symmetric non-orientable genus, 227 vertex-accumulation point, 303 syntheme, 262 vertex-homogeneous map, 310 vertex-set, 1 vertex-transitive graph, 2 Tait colouring, 323 Vizing’s theorem, 9

Index 347

voltage, 27 weight of walk, 331 voltage assignment, 182 well-quasi-order, 82 voltage graph, 26, 27 width of graph, 117 VSLI design, 63

Xuong tree, 36 Wagner’s conjecture, 82 Wagner’s theorem, 11 walk, 3 weak homogeneous representation, 176 Y- ordering, 318 weak negative support, 157 Y -exchange, 72 weak nodal domain, 157 weak positive support, 157 weak realizability problem, 145 zip product of graphs, 142