Here - Combinatorial algorithms and algorithmic graph theory

More documents

Recommendations

Info

28 Generation of cubic graphs |V (G)| # prime graphs # cubic graphs 4 1 1 6 0 2 8 1 5 10 1 19 12 1 85 14 3 509 16 2 4 060 18 5 41 301 20 4 510 489 22 9 7 319 447 24 11 117 940 535 26 16 2 094 480 864 28 32 40 497 138 011 30 37 845 480 228 069 32 73 18 941 522 184 590 Table 2.1: Number of prime graphs vs. number of cubic graphs. 2.4 Generation of graphs with reducible triangles 2.4.1 Introduction As can be seen from Table 2.1, the number of prime graphs is very small compared to the number of cubic graphs. So the efficiency of the algorithm is entirely determined by the efficiency of the algorithm to generate non-prime graphs. These graphs are constructed by the edge insertion operation from Figure 2.9. The class of connected cubic graphs is closed under the edge insertion operation, so the only time consuming routines are those that make sure that only pairwise non-isomorphic graphs are generated. As with the generation for prime graphs, we also use the canonical construction path method for this. When using the canonical construction path method, we first have to define a canonical reduction which is unique up to isomorphism for every non-prime graph. The inverse of a canonical reduction is a canonical expansion. Recall from Section 1.3 that the two rules of this method are: 1. Only accept a graph if it was constructed by a canonical expansion.
2.4. Generation of graphs with reducible triangles 29 2. For every graph G to which construction operations are applied, only perform one expansion from each equivalence class of expansions of G. The isomorphism rejection for non-prime graphs is described in more detail in Sections 2.4.2 and 2.5.1. Figure 2.9: The basic edge insertion operation. The theory of random cubic graphs (see [122] for example) says that the number of triangles is asymptotically a Poisson distribution with mean 4 3 . So asymptotically the probability that a cubic graph contains no triangles is e − 4 3 = 0.264... The expected number of copies of K4 − is o(1), so this implies that asymptotically about 74% of cubic graphs have triangles whose edges are reducible (that are triangles which are not contained in a K4 − ). This percentage is also approximately true for the graphs of the small sizes we are dealing with, which justifies paying special attention to these reducible triangles as they allow some optimisations that do not work in general. Table 2.2 shows statistics about the number of reducible triangles in cubic graphs with 26 vertices. # reducible # graphs triangles 0 497 010 000 1 774 885 044 2 540 977 972 3 218 274 256 4 54 459 966 5 8 183 373 6 666 137 7 23 837 8 279 Table 2.2: Counts of all 2 094 480 864 connected cubic graphs with 26 vertices according to the number of reducible triangles they have.
Page 1: Ghent University Faculty of Science
Page 5 and 6: Dankwoord Deze thesis zou er zeker
Page 7 and 8: Contents Summary vii 1 Introduction
Page 9 and 10: Contents v 5.2.1 Introduction . . .
Page 11 and 12: Summary In this thesis we develop e
Page 13 and 14: Summary ix represent carbon atoms.
Page 15 and 16: Chapter 1 Introduction A graph is a
Page 17 and 18: 1.1. Definitions and preliminaries
Page 19 and 20: 1.2. Exhaustive generation 5 A k-fa
Page 21 and 22: 1.3. Isomorphism-free generation 7
Page 23 and 24: 1.3. Isomorphism-free generation 9
Page 25 and 26: Chapter 2 Generation of cubic graph
Page 27 and 28: 2.2. The generation algorithm 13 of
Page 29 and 30: 2.3. Generation of prime graphs 15
Page 41: 2.3. Generation of prime graphs 27
Page 45 and 46: 2.4. Generation of graphs with redu
Page 47 and 48: 2.4. Generation of graphs with redu
Page 49 and 50: 2.5. Generation of non-prime graphs
Page 57 and 58: 2.6. Generation of graphs with girt
Page 59 and 60: 2.6. Generation of graphs with girt
Page 61 and 62: 2.7. Generation of graphs with conn
Page 63 and 64: 2.7. Generation of graphs with conn
Page 65 and 66: 2.9. Closing remarks 51 strongly on
Page 67 and 68: 2.9. Closing remarks 53 All connect
Page 69 and 70: Chapter 3 Generation of snarks In t
Page 71 and 72: 3.1. Introduction 57 smallest snark
Page 73 and 74: 3.3. The generation algorithm 59 As
Page 75 and 76: 3.4. Optimisations 61 At first sigh
Page 77 and 78: 3.4. Optimisations 63 the non-isomo
Page 79 and 80: 3.5 Testing and results 3.5. Testin
Page 81 and 82: 3.5. Testing and results 67 Order 1
Page 83 and 84: 3.6. Closing remarks 69 Figure 3.2:
Page 85 and 86: 3.6. Closing remarks 71 snarks of s
Page 87 and 88: Chapter 4 Generation of fullerenes
Page 89 and 90: 4.1. Introduction 75 However, Whitn
Page 91 and 92: 4.1. Introduction 77 incomplete in
Page 93 and 94:
4.2. Generation of fullerenes 79 or
Page 95 and 96:
4.2. Generation of fullerenes 81 C
Page 97 and 98:
4.2. Generation of fullerenes 83 00
Page 99 and 100:
4.2. Generation of fullerenes 85 wi
Page 101 and 102:
4.2. Generation of fullerenes 87 We
Page 103 and 104:
4.2. Generation of fullerenes 89 e
Page 105 and 106:
4.3. Generation of IPR fullerenes 9
Page 107 and 108:
Page 109 and 110:
Page 111 and 112:
Page 113 and 114:
Page 115 and 116:
Page 117 and 118:
Page 119 and 120:
Page 121 and 122:
Page 123 and 124:
Page 125 and 126:
Page 127 and 128:
4.4. Testing and results 113 adding
Page 129 and 130:
4.4. Testing and results 115 number
Page 131 and 132:
4.4. Testing and results 117 nv nf
Page 133 and 134:
Page 135 and 136:
Page 137 and 138:
4.4. Testing and results 123 Figure
Page 139 and 140:
4.5. Closing remarks 125 Figure 4.3
Page 141 and 142:
Chapter 5 Ramsey numbers In the fir
Page 143 and 144:
5.2. Generalised triangle Ramsey nu
Page 145 and 146:
Page 147 and 148:
Page 149 and 150:
Page 151 and 152:
Page 153 and 154:
Page 155 and 156:
Page 157 and 158:
Page 159 and 160:
Page 161 and 162:
Page 163 and 164:
Page 165 and 166:
5.3. Classical triangle Ramsey numb
Page 167 and 168:
Page 169 and 170:
Page 171 and 172:
Page 173 and 174:
Page 175 and 176:
Page 177 and 178:
Page 179 and 180:
Page 181 and 182:
Page 183 and 184:
Page 185 and 186:
Page 187 and 188:
Page 189 and 190:
Page 191 and 192:
Page 193 and 194:
Page 195 and 196:
Page 197 and 198:
Appendix A Notation 183
Page 199 and 200:
Appendix B Ramsey numbers of connec
Page 201 and 202:
187 and contains one of the graphs:
Page 203 and 204:
Appendix C Number of Ramsey graphs
Page 205 and 206:
191 edges number of vertices n e 19
Page 207 and 208:
193 edges number of vertices n e 29
Page 209 and 210:
Bibliography [1] E. Albertazzi, C.
Page 211 and 212:
Bibliography 197 [23] G. Brinkmann
Page 213 and 214:
Bibliography 199 [48] G. Exoo. On t
Page 215 and 216:
Bibliography 201 [74] C. Justus. Tr
Page 217 and 218:
Bibliography 203 [101] S.P. Radzisz
Page 219 and 220:
Bibliography 205 [126] C.Q. Zhang.
Page 221 and 222:
Nederlandstalige samenvatting In de
Page 223 and 224:
Nederlandstalige samenvatting 209 v
Page 225 and 226:
List of Figures 1.1 The basic edge
Page 227 and 228:
List of Figures 213 4.13 An L 2 exp
Page 229 and 230:
List of Tables 2.1 Number of prime
Page 231 and 232:
List of Tables 217 5.2 Counts and g
Page 233 and 234:
Index 219 edge-cut, 4 embedding, 74
show all

Here - Combinatorial algorithms and algorithmic graph theory

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?