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16 Generation of cubic graphs reduced graph is a connected cubic graph without parallel edges or loops. Note that each reducible non-adjacent edge diamond is also a reducible edge diamond. Figure 2.5: The construction operations for prime graphs. To prove that all prime graphs can be generated by our construction operations, we will show in Section 2.3.2 that each prime graph can be reduced to a smaller prime graph by one of the reduction operations (i.e. Lemma 2.2). Note that the class of prime graphs is not closed under these construction operations (for an example, see Figure 2.6). So after applying an operation it must be tested whether the new graph is a prime graph or not. We use some simple look-aheads that avoid the construction of non-prime graphs. These look-aheads and other details about the construction are explained in Section 2.3.3. In Section 2.3.2 we explain how we make sure that the algorithm does not output isomorphic prime graphs and prove that our algorithm generates exactly one representative of every isomorphism class of prime graphs. 2.3.2 Isomorphism rejection We use the canonical construction path method (see Section 1.3) to make sure no isomorphic prime graphs are generated. We will now describe how to apply the canonical construction path method for the generation of prime graphs. Note that the lollipop insertion operation and the edge diamond insertion operation are applied to edges, while the non-adjacent edge diamond insertion operation is applied to pairs of non-adjacent edges. Two lollipop (or edge diamond) expansions of a prime graph G are equivalent if there is an automorphism
2.3. Generation of prime graphs 17 Figure 2.6: Application of the edge insertion operation which yields a non-prime graph. of G mapping the edges onto each other (i.e. they are in the same orbit of edges of the automorphism group of G). Two non-adjacent edge diamond expansions are equivalent if there is an automorphism of G mapping the pairs of edges onto each other. Recall from Section 1.3 that we first have to define a canonical reduction which is unique up to isomorphism and that the two rules of the canonical construction path method (applied to the generation of prime graphs) are: 1. Only accept a prime graph if it was constructed by a canonical expansion. 2. For every prime graph G to which construction operations are applied, only perform one expansion from each equivalence class of expansions of G. A canonical expansion is the inverse of a canonical reduction. The three construction operations for prime graphs were introduced in Section 2.3.1. We will now introduce the concepts of canonical reducible lollipops, edge diamonds and non-adjacent edge diamonds in order to define a canonical reduction. The concepts of central vertex, extremal vertex and reducible lollipop, edge diamond and non-adjacent edge diamond were already introduced in Section 2.3.1 and Aut(G) stands for the automorphism group of a graph G. Definition 2.1 (Canonical reducible lollipop). Given a graph G and a canonical representative function f. We call a reducible lollipop of G canonical if it is in the same orbit of Aut(G) as the reducible lollipop which contains the central vertex with the largest label in f(G) (i.e. the canonically labelled representative of G).
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: 2.3. Generation of prime graphs 15
Page 33 and 34: 2.3. Generation of prime graphs 19
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 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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