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12 Generation of cubic graphs (a) (b) (c) Figure 2.1: Some examples of cubic graphs: K 4, the Petersen graph and the C 60 fullerene respectively. The class of cubic graphs is especially interesting for mathematical applications because for various important open problems in graph theory, the smallest or simplest possible potential counterexamples are cubic graphs (see Chapter 3 for more information). In chemistry, cubic graphs serve as models for e.g. the Nobel Prize winning fullerenes [80] (see Chapter 4) or, more generally, for some cyclopolyenes [5]. The generation of cubic graphs can be considered a benchmark problem in structure enumeration. The first complete lists of cubic connected graphs were given by de Vries at the end of the 19th century, who gave a list of all cubic (connected) graphs up to 10 vertices [42, 43]. The first computer approach was by Balaban, a theoretical chemist. He generated all cubic graphs up to 12 vertices in 1966/67 [5]. De Vries’ lists were independently confirmed by hand by Bussemaker and Seidel in 1968 [38] and Imrich in 1971 [70]. From 1974 on, various algorithms for the generation of cubic graphs were published. Each algorithm was implemented in a computer program that could generate larger lists of cubic graphs, see [97],[50],[37],[92],[11]. In 1983 Robinson and Wormald [106] published a paper on the non-constructive enumeration of cubic graphs. When present research began, the fastest publicly available program for the generation of cubic graphs was minibaum [11]. When developed in 1992, minibaum could be used to generate complete lists of all cubic graphs up to 24 vertices and several more restricted subclasses of cubic graphs with more vertices, like cubic bipartite graphs or cubic graphs with higher girth. Later, when more and faster computers were available, minibaum was used to generate all cubic graphs up to 30 vertices in order to test them for Yutsis decompositions [2]. In 1999, Meringer [95] published a very efficient algorithm for the generation
2.2. The generation algorithm 13 of regular graphs of given degree. But for the generation of all cubic graphs the program based on his algorithm is slower than minibaum. In 2000, Sanjmyatav [110] and her supervisor McKay developed a set of very fast specialised programs for various classes of cubic graphs. Unfortunately these programs were never released or published. In this chapter we describe an algorithm based on ideas already used by de Vries and Sanjmyatav/McKay but also with several new ideas. Our implementation of this algorithm is faster than any previous program. The algorithm is described in Sections 2.2-2.5. In Section 2.6 we describe how we adapted the algorithm to generate cubic graphs with a non-trivial lower bound on the girth efficiently and in Section 2.7 we describe how the algorithm can be used to generate cubic graphs with certain connectivity requirements. We conclude this chapter with a comparison of our algorithm with previous generators for cubic graphs. 2.2 The generation algorithm Our basic construction operation to make a cubic graph G ′ with n vertices from a cubic graph G with n − 2 vertices is the insertion of a new edge between new vertices inserted in two different edges of G. This operation was already used by de Vries and can be seen in Figure 2.2. The inverse operation involves removing an edge {x, y} and its endpoints x and y, then adding an edge between the two vertices other than y that were previously adjacent to x, and an edge between the two vertices other than x that were previously adjacent to y. We call this an edge reduction operation in case the resulting graph is a connected cubic graph. In that case we call the edge e = {x, y} reducible, otherwise irreducible. So e is irreducible if and only if it is a bridge, has an endpoint in a triangle that does not contain e, or has two endpoints in the same 4-gon that does not contain e. The latter two situations are depicted in Figure 2.3. Figure 2.2: The basic edge insertion operation. A cubic connected graph without reducible edges is called a prime graph. By definition, each connected cubic graph can be constructed from a prime graph by
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: Chapter 2 Generation of cubic graph
Page 29 and 30: 2.3. Generation of prime graphs 15
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
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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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