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24 Generation of cubic graphs Figure 2.7: A non-adjacent edge diamond insertion which yields a new reducible lollipop with central vertex c. The non-adjacent edge diamond insertion operation is applied to pairs of nonadjacent edges. Suppose that we have a prime graph G, then the following cases can occur: • G contains at least 3 reducible lollipops. The non-adjacent edge diamond insertion operation can destroy at most 2 reducible lollipops. So the list of non-adjacent edges which are eligible for expansion is empty. • G contains contains 2 reducible lollipops. The lollipops can only be destroyed if both endpoints of the first edge in the pair of edges to which the expansion is applied are in the K 4 + of the first reducible lollipop and the endpoints of the second edge of the edge pair are in the K 4 + of the second reducible lollipop. However applying the nonadjacent edge diamond operation to such a pair of edges yields a non-prime graph. So also in this case the list of non-adjacent edges which are eligible for expansion is empty.
2.3. Generation of prime graphs 25 • G contains 1 reducible lollipop. The reducible lollipop can only be destroyed if both endpoints of an edge in the pair of edges to which the non-adjacent edge diamond expansion is applied are in the K 4 + of the reducible lollipop. However applying an expansion to such a pair of edges yields a non-prime graph. So also here the list of non-adjacent edges which are eligible for expansion is empty. • G contains no lollipops. Here we only have to make sure we do not generate any new reducible lollipops and that the expanded graph is still prime. The pairs of edges which can be chosen for expansion consist of all pairs of non-adjacent edges where none of the edges is a diamond edge. When an expansion is applied we still have to check if no new reducible lollipops are generated, because the non-adjacent diamond edge insertion operation can turn a K4 − into a reducible lollipop (cf. Figure 2.7). We also have to test if the expanded graphs are prime. For the efficiency of the algorithm it is not a problem that various graphs might be rejected by the generation algorithm for prime graphs. Since, as is shown in Section 2.3.4, the generation of prime graphs is certainly not the bottleneck in the generation algorithm for all cubic graphs. Edge diamond insertion operation Since the edge diamond operation has the lowest priority among the three operations for prime graphs, we have to make sure that this insertion operation destroys all existing reducible lollipops and reducible non-adjacent edge diamonds without generating any new ones. Next to that, we also have to make sure that the expanded graphs are still prime. The edge diamond insertion operation is applied to edges. Suppose that we have a prime graph G, the following cases can occur: • G contains 2 reducible lollipops and no reducible non-adjacent edge diamonds. It is only possible to destroy these 2 reducible lollipops if the 2 central vertices of those lollipops are neighbours. This is only the case in the graph of Figure 2.8. Here the only edge which is eligible for expansion is the edge connecting the 2 central vertices.
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 37: 2.3. Generation of prime graphs 23
Page 43 and 44: 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
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4.1. Introduction 75 However, Whitn
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4.1. Introduction 77 incomplete in
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4.2. Generation of fullerenes 79 or
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4.2. Generation of fullerenes 81 C
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4.2. Generation of fullerenes 83 00
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4.2. Generation of fullerenes 85 wi
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4.2. Generation of fullerenes 87 We
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4.2. Generation of fullerenes 89 e
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4.3. Generation of IPR fullerenes 9
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4.4. Testing and results 113 adding
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4.4. Testing and results 115 number
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4.4. Testing and results 117 nv nf
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4.4. Testing and results 123 Figure
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4.5. Closing remarks 125 Figure 4.3
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Chapter 5 Ramsey numbers In the fir
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5.2. Generalised triangle Ramsey nu
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5.3. Classical triangle Ramsey numb
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Appendix A Notation 183
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Appendix B Ramsey numbers of connec
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187 and contains one of the graphs:
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Appendix C Number of Ramsey graphs
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191 edges number of vertices n e 19
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193 edges number of vertices n e 29
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Bibliography [1] E. Albertazzi, C.
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Bibliography 197 [23] G. Brinkmann
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Bibliography 199 [48] G. Exoo. On t
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Bibliography 201 [74] C. Justus. Tr
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Bibliography 203 [101] S.P. Radzisz
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Bibliography 205 [126] C.Q. Zhang.
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Nederlandstalige samenvatting In de
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Nederlandstalige samenvatting 209 v
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List of Figures 1.1 The basic edge
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List of Figures 213 4.13 An L 2 exp
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List of Tables 2.1 Number of prime
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List of Tables 217 5.2 Counts and g
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Index 219 edge-cut, 4 embedding, 74
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