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2 Introduction structure elucidation, the goal is to identify molecules. By analysing the mass spectrum of a substance, chemists can determine the chemical formula of the molecule of which the substance consists together with forbidden or forced substructures. One can then use a program to generate all structures which represent molecules with this chemical formula and forbidden/forced substructures and compute the spectrum of the generated structures. Then by comparing the computed spectra with the measured spectrum of the substance one can often propose a small list of candidate structures. This helps to determine the structure of the molecule. The DENDRAL project [82] which was initiated in the sixties at Stanford University was the first project for the automatic recognition of the structure of molecules. In structure prediction, a generation algorithm is used to generate all graphs which represent molecules of a given class which have useful properties. One can then compute the chemical energy of each of the generated structures. The ones with the best energy are chemically the most stable ones and chemists can investigate their potential applications and the possibility to synthesise these molecules. Programs for generating models of fullerenes (see Chapter 4) have also been used by chemists to investigate the hypothesis that IPR fullerenes are chemically more stable than non-IPR fullerenes [1]. Of course structure generation has also applications outside the field of mathematics or chemistry. Structure generation algorithms have for example already been used to determine the optimal pitch sequence of tires. The surface of a tire can contain several types of blocks, e.g. short, medium and long ones. The sequence of these blocks is called the pitch sequence of the tire. One pitch sequence can be more resistant to wear or have a lower noise level than another. So in order to determine the best pitch sequence, one can generate all possible pitch sequences (here the generated structures are not graphs) and compute the resistance to wear or noise level of each generated pitch sequence. In this thesis we develop algorithms for the generation of complete lists of graphs which have important applications in mathematics as well as generation algorithms which are important in chemistry. In principle one can generate all graphs from a specific graph class by using an existing algorithm to generate all graphs with a filter at the end which only outputs graphs which are part of the specific graph class. However for most graph classes this approach is very inefficient as often only a tiny percentage of all graphs are part of this graph class. For example only 0.000000057 percent of
1.1. Definitions and preliminaries 3 the graphs with 12 vertices are cubic and the percentage is decreasing rapidly. So for interesting graph classes it is justified to develop specialised algorithms to generate graphs from these classes efficiently. Note: several parts of this thesis were taken from articles which were written in collaboration with not only the promotor of this thesis, but also with other co-authors. Parts which were mainly developed and written by other co-authors are clearly marked. 1.1 Definitions and preliminaries In this section we give the definitions and notations which are used in this thesis. Most of them are widely used in the field of graph theory. For a good introduction to graph theory, we refer the reader to [44] or [120]. Unless stated otherwise, all graphs in this thesis are simple and undirected. Let G be such a graph. The vertex set of G is denoted by V (G) and the edge set of G by E(G). Sometimes we also denote a graph as G = (V, E). Here V represents the set of vertices of G and E the set of edges. We also refer to |V | as the order of G. Two vertices v, w ∈ V (G) are called adjacent if {v, w} ∈ E(G). We say that an edge e ∈ E(G) is incident to v ∈ V (G) if v ∈ e. Two edges are called adjacent if they share a vertex. The degree of a vertex v ∈ V (G) is the number of edges which are incident to v and is denoted by deg G (v) (or simply deg(v) if G is fixed). The minimum and maximum degree of vertices in G is denoted by δ(G) and ∆(G), respectively. If δ(G) = ∆(G) = d, G is called d-regular. The set of neighbours (i.e. adjacent vertices) of a vertex v ∈ V (G) is written as N v (G) (or N(v) if G is fixed). A graph G ′ is a subgraph of G if V (G ′ ) ⊆ V (G) and E(G ′ ) ⊆ E(G). If G ′ is a subgraph of G and V (G ′ ) = V (G), then G ′ is a spanning subgraph of G. If G ′ is a subgraph of G and ∀ v, w ∈ V (G ′ ) the following holds: {v, w} ∈ E(G) ⇒ {v, w} ∈ E(G ′ ), then G ′ is called an induced subgraph of G. We also refer to G ′ as the subgraph of G induced by the set of vertices X = V (G ′ ) (denoted by G[X]). If G ′ is a subgraph of G, then G is also called a supergraph of G ′ . The complement of a graph G = (V, E) (denoted by G c ), is the graph G c = (V, ( V 2) \ E). A subgraph P of a graph G is a path if V (P ) = {v 0 , v 1 , ..., v n } and E(P ) = {{v i , v i+1 } | 0 ≤ i ≤ n} where all vertices of V (P ) are distinct. The length of 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: Chapter 1 Introduction A graph is a
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 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
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2.9. Closing remarks 53 All connect
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Chapter 3 Generation of snarks In t
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3.1. Introduction 57 smallest snark
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3.3. The generation algorithm 59 As
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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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