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14 Generation of cubic graphs (a) (b) Figure 2.3: Two examples of an irreducible edge e: an edge which has an endpoint in a triangle that does not contain e (i.e. Figure 2.3a) and an edge with two endpoints in the same 4-gon that does not contain e (i.e. Figure 2.3b). recursive application of the edge insertion operation. The class of all connected cubic graphs is constructed in two steps: K 4 ⇓ (operations to generate prime graphs) Prime graphs ⇓ (edge insertion operation) All cubic graphs The construction operations to generate prime graphs are described in Section 2.3. The second step is described in more detail in Sections 2.4 and 2.5. In both steps we use the canonical construction path method (see Section 1.3) to make sure no isomorphic graphs are output by the algorithm. In those sections we also describe how this isomorphism rejection technique is applied for the generation of cubic graphs. 2.3 Generation of prime graphs 2.3.1 Introduction We refer to a subgraph of a graph isomorphic to K 4 − e (with e an edge of K 4 ) as a K4 − . The two vertices in a K− 4 that have degree 2 in this subgraph are called extremal vertices. This is shown in Figure 2.4a, where the extremal vertices are labelled v 1 and v 2 . K 4 with an edge subdivided by inserting a vertex of degree 2 is referred to as K 4 + . The vertex in a K+ 4 which has degree 2 in this subgraph is
2.3. Generation of prime graphs 15 called the central vertex. This is shown in Figure 2.4b where the central vertex is labelled v c . (a) (b) Figure 2.4: A K − 4 and a K+ 4 Lemma 2.1. A connected cubic graph is a prime graph if and only if each edge is a bridge or has an endpoint in a K − 4 . Proof. The direction ⇐ is immediate, because neither bridges nor edges with endpoints in a K4 − can be reduced. So suppose that e = {x, y} is an edge in a prime graph G which is not a bridge and has no endpoints in a K4 − . Note that if an edge has two endpoints in the same cycle, the edge reduction applied to this edge can reduce the length of the cycle by at most 2. If the endpoints of the reduced edge are not part of the same cycle, the edge reduction can reduce the length of each cycle in which the reduced edge has endpoints by at most 1. Also note that if an edge in a prime graph has two endpoints in the same square, it is part of a K4 − . Thus if e has no endpoints in a triangle, then e is reducible. On the other hand, if x is contained in a triangle, but not in a K4 − , each edge of the triangle is reducible. We use the three construction operations from Figure 2.5 to generate all prime graphs. We also refer to operation (a) as the lollipop insertion operation, operation (b) as the edge diamond insertion operation and operation (c) as the non-adjacent edge diamond insertion operation. We call a K 4 + which is part of a prime graph and which is reducible by a lollipop reduction a reducible lollipop. A K4 − which is part of a prime graph and which is reducible by an edge diamond reduction a reducible edge diamond and similarly a K4 − which is reducible by a non-adjacent edge diamond reduction is called a reducible non-adjacent edge diamond. Reductions are only valid if the
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
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Page 25 and 26: Chapter 2 Generation of cubic graph
Page 27: 2.2. The generation algorithm 13 of
Page 31 and 32: 2.3. Generation of prime graphs 17
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
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Page 61 and 62: 2.7. Generation of graphs with conn
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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
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3.5 Testing and results 3.5. Testin
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3.5. Testing and results 67 Order 1
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3.6. Closing remarks 69 Figure 3.2:
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3.6. Closing remarks 71 snarks of s
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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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