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4 Introduction path is its number of edges. The distance d(v, w) between 2 vertices v, w ∈ V (G) is the length of the shortest path from v to w in G. If there is no path from v to w, d(v, w) is defined as ∞. The diameter of G is the greatest distance between any two vertices in G. A subgraph C of a graph G is a cycle if V (C) = {v 0 , v 1 , ..., v n } and E(C) = {{v i , v i+1 } | 0 ≤ i < n} ∪ {v 0 , v n } where all vertices of V (C) are distinct. The length of the cycle is its number of edges (or vertices). A complete graph of order n (denoted K n ) is a graph G = (V, E) with E = ( V 2) . A graph G is called bipartite if there are two sets of vertices V 1 and V 2 such that V 1 ∩ V 2 = ∅ and V 1 ∪ V 2 = V (G) and ∀ e ∈ E(G) : |e ∩ V 1 | = |e ∩ V 2 | = 1. A complete bipartite graph G = (V 1 ∪V 2 , E) is a bipartite graph with partitions V 1 and V 2 such that for any two vertices v 1 ∈ V 1 and v 2 ∈ V 2 , {v 1 , v 2 } is an edge in G. The complete bipartite graph with partitions of size |V 1 | = m and |V 2 | = n, is denoted K m,n . The girth of a graph G is the length of its shortest cycle and is denoted by g(G). For acyclic graphs the girth is defined as ∞. A tree is a connected acyclic graph. A graph G = (V, E) is connected if for every v, w ∈ V there is a path from v to w. A disconnected graph is a graph which is not connected. Given a connected graph G = (V, E). A set S ⊆ V is a vertex-cut if G[V \ S] is disconnected. If the vertex-cut only contains one vertex, we call that vertex a cutvertex. A graph G is k-connected if G is connected and no vertex-cut S exists with |S| < k. Given a connected graph G = (V, E). A set S ⊆ E is an edge-cut if the graph G ′ = (V, E \ S) is disconnected. If the edge cut only contains one edge, we call that edge a bridge. A graph G is k-edge-connected if G is connected and no edge-cut S exists with |S| < k. We call an (edge) cut S with |S| = k a k-(edge)- cut. Thus if a graph is k-edge-connected but not (k + 1)-edge-connected, it has a k-edge-cut. A graph G is cyclically k-edge-connected if the deletion of fewer than k edges from G does not create two components both of which contain at least one cycle. The largest integer k such that G is cyclically k-edge-connected is called the cyclic edge-connectity of G and is denoted by λ c (G). If no set of edges can be deleted such that two components are created which both contain at least one cycle, the cyclic edge-connectivity is defined as ∞ (so λ c (K 4 ) = ∞). The chromatic index χ ′ (G) of a graph G is the minimum number of colours required for an edge colouring of that graph such that no two adjacent edges have the same colour.
1.2. Exhaustive generation 5 A k-factor of a graph G is a spanning k-regular subgraph of G. A 1-factor is also called a perfect matching. A graph is called hamiltonian if it contains a hamiltonian cycle, this is a cycle which contains all vertices of the graph. An independent set of G is a set of vertices S ⊆ V (G) such that ∀ v, w ∈ S : {v, w} /∈ E(G). The size of the largest independent set of G is called the independence number of G and is denoted by α(G). Two graphs G and G ′ are isomorphic (denoted by G ∼ = G ′ ) if and only if there is a bijective function φ : V (G) → V (G ′ ) : {v, w} ∈ E(G) ⇐⇒ {φ(v), φ(w)} ∈ E(G ′ ). The function φ is called an isomorphism from G to G ′ . An isomorphism from G to itself is called an automorphism. The automorphism group Aut(G) of G is the group of all automorphisms of G. If the automorphism group only contains one element (namely the identity automorphism which maps every vertex to itself), we call that group trivial. The orbit of a vertex v ∈ V (G) is the set of vertices to which v can be mapped by an automorphism of G. In this thesis we also use several definitions which are specific for plane graphs. These definitions are introduced in Chapter 4. For an overview of the notations which are used in this thesis, see Appendix A. 1.2 Exhaustive generation All algorithms described in this thesis are designed to generate graphs from a specific class of graphs (e.g. cubic graphs, fullerenes,...) efficiently. All of these algorithms generate these classes of graphs exhaustively, i.e. they generate all graphs of a given graph class (for a given number of vertices). The graphs are obtained by recursively applying construction operations to a set of initial graphs. A construction or expansion operation is an operation which constructs a larger graph from a given graph. A reduction operation is an operation inverse to a construction operation. We call the graphs which are obtained by applying a construction operation to a smaller graph children and this smaller graph where these children are obtained from the parent. We also call the graph obtained by applying an expansion to a graph an expanded graph. To prove that all graphs of a given class can be generated by our algorithm, we usually prove that each graph from the class of graphs (except for a small number of irreducible graphs) can be reduced to a smaller graph of the same class by our reduction operations.
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: 1.1. Definitions and preliminaries
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
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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
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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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