Generation of Graph Classes with Efficient Isomorph Rejection S. Narjess Afzaly A thesis submitted for the degree of Doctor of Philosophy of The Australian National University August 2016 c S. Narjess Afzaly 2016 Declaration Except where otherwise indicated, this thesis is my own original work. S. Narjess Afzaly 25 August 2016 iii To my dearest grandmother, Ashraf (Seyedeh Masoumeh GhadimiNejhad) Acknowledgments Foremost, I would like to express my sincere gratitude to my supervisor Professor Brendan McKay for the continuous support of my PhD study and research, for his pa- tience, care, enthusiasm and immense knowledge. I am extremely grateful for every- thing I learned from him and for the insight I gained working with him. In addition, I would like to thank the rest of my thesis committee members, Professor Weifa Liang, Dr Pascal Schweitzer, and especially Dr Scott Morrison for the guidance, for what I have learned working with him, and for his enthusiasm. My acknowledgment also goes to Professor Ian Wanless, Dr Catherine Greenhill, Professor Tomasz Luczak, Professor Piotr Sankowski and Professor Andrzej Ruciñski for providing me the opportunity to visit their respective departments. I am grateful to Dr Beáta Faller, Mr Michał D ˛ebski,Dr David Penneys, Dr Vikram Kamat, Professor Stanisław Radziszowski, Dr Erika Skrabulakova, Dr Joanna Polcyn-Lewandowska and Professor Vladimir Modrák, for their insightful discussion. I thank my parents, Ali and Nahid, my grandmother, Ashraf, and my two broth- ers, HamidReza and MahmoudReza, for supporting me throughout my life and en- couraging me to take the path of knowledge. I also thank my friends Sanam, Fiona, Ian, Bishan, Anila, Rani, Sahar, David, Pierre, Sara, Richa, Quifen, Edward, Samuel, Khoi-Nguyen, Ehsan, Azadeh, Jessie, Soheila, Soraya and other friends at CSIT and University House for their priceless support. I appreciate the services ANU offers to students especially through University House, PARSA, CHELT, Counselling Centre and ANU Security. I am thankful for the support and services provided by the great staff members at CSIT, especially Mrs Janette Rawlinson. I also acknowledge the places where I wrote a big portion of my thesis: Telstra tower, Max Brenner, Grill’d, YHA, and ANU Canberra and Kioloa Coastal Campus in Australia, Cafe Artist, Cafe Lemon and Hotel Shabestan in Rasht, and the trains around Poland. vii Abstract In this thesis, efficient isomorph-free generation of graph classes with the method of generation by canonical construction path(GCCP) is discussed. The method GCCP has been invented by McKay in the 1980s. It is a general method to recursively gener- ate combinatorial objects avoiding isomorphic copies. In the introduction chapter, the method of GCCP is discussed and is compared to other well-known methods of gen- eration. The generation of the class of quartic graphs is used as an example to explain this method. Quartic graphs are simple regular graphs of degree four. The programs, we developed based on GCCP, generate quartic graphs with 18 vertices more than two times as efficiently as the well-known software GENREG does. This thesis also demonstrates how the class of principal graph pairs can be gener- ated exhaustively in an efficient way using the method of GCCP. The definition and importance of principal graph pairs come from the theory of subfactors where each subfactor can be modelled as a principal graph pair. The theory of subfactors has applications in the theory of von Neumann algebras, operator algebras, quantum al- gebras and Knot theory as well as in design of quantum computers. While it was initially expected that the classification at index 3 + p5 would be very complicated, using GCCP to exhaustively generate principal graph pairs was critical in completing 1 the classification of small index subfactors to index 5 4 . The other set of classes of graphs considered in this thesis contains graphs without a given set of cycles. For a given set of graphs, , the Turán Number of , ex(n, ), H H H is defined to be the maximum number of edges in a graph on n vertices without a subgraph isomorphic to any graph in . Denote by EX(n, ), the set of all extremal H H graphs with respect to n and , i.e., graphs with n vertices, ex(n, ) edges and no H H subgraph isomorphic to any graph in . We consider this problem when is a set of H H cycles. New results for ex(n, ) and EX(n, ) are introduced using a set of algorithms C C based on the method of GCCP. Let be an arbitrary subset of C , C , C ,..., C . K f 3 4 5 32g For given n and a set of cycles, , these algorithms can be used to calculate ex(n, ) C C and extremal graphs in Ex(n, ) by recursively extending smaller graphs without any C cycle in where = K or = C , C , C ,... and n 64. These results are C C C f 3 5 7 g [ K ≤ considerably in excess of the previous results of the many researchers who worked on similar problems. ix x In the last chapter, a new class of canonical relabellings for graphs, hierarchical canonical labelling, is introduced in which if the vertices of a graph, G, is canonically labelled by 1, . , n , then G n is also canonically labelled. An efficient hierarchi- f g n f g cal canonical labelling is presented and the application of this labelling in generation of combinatorial objects is discussed. Contents Acknowledgments vii Abstract ix 1 Introduction 1 1.1 Definitions and Preliminaries . 3 1.2 Methods of Generation . 5 1.2.1 The Method of Naive Generation . 5 1.2.2 Generation by Canonical Representatives (Orderly Generation) . 6 1.2.3 Generation by Canonical Construction Path . 6 1.3 GCCP and Generation of Quartic Graphs . 8 1.3.1 How Isomorphic Copies are Avoided . 8 1.3.2 Generation of Quartic Graphs . 9 1.3.2.1 Results . 15 1.3.3 Variations of GCCP . 17 1.3.4 Parallelisation in GCCP . 17 1.4 Orderly Generation Versus GCCP . 17 1.5 Nauty; The Automorphism Group and a Canonical Labelling . 19 1.6 Overview of the Thesis . 20 2 Exhaustive Generation of Principal Graph Pairs 23 2.1 Abstract . 23 2.2 Introduction . 23 2.2.1 Subfactors and Principal Graph Pairs . 24 2.3 Preliminaries . 25 2.3.1 Definitions . 25 2.3.2 Data Structures and Notations . 27 2.3.2.1 Using Sparse Graph Data Structure to Store PGPs . 27 2.4 The Generation Algorithm . 28 2.4.1 Extension . 31 2.4.2 Reduction . 32 2.4.3 Genuine Reduction . 32 xi xii Contents 2.4.4 Pseudo-Code . 36 2.5 Efficiency . 40 2.5.1 Using Common Computation . 40 2.5.2 Efficient Data Structure . 40 2.5.3 Group Calculations . 42 2.5.3.1 Calculating the Canonical Labelling . 42 2.5.3.2 Avoiding Equivalent Extensions of a PGP . 43 2.5.4 Associativity . 46 2.5.5 Index of PGPs: Calculation and Pruning Techniques . 51 2.5.5.1 Calculating Lower Bounds for Index of a PGP Using Power Iteration Method . 52 2.5.5.2 Efficient Calculations of the Maximum Eigenvalue of a Bipartite Graph . 53 2.5.5.3 Upper Bound on the Degree of Vertices . 54 2.5.6 Giving Priority to One Graph of a PGP . 58 2.6 Formal Proof . 60 2.7 The C Code Program . 64 2.7.1 The Features maxtime and resume . 65 2.8 Conclusion . 66 3 The Turán Numbers for Cycles 69 3.1 Abstract . 69 3.2 Introduction . 69 3.2.1 Definitions and Preliminaries . 70 3.3 Literature Review . 71 3.3.1 Turán Numbers for C4 ......................... 71 3.3.1.1 Some Bounds and Asymptotic Behavior . 72 3.3.2 Turán Numbers for C , C ..................... 74 f 3 4g 3.3.3 Turán Numbers for C , C ..................... 74 f 4 5g 3.3.4 Turán Numbers for C , C , C ................... 74 f 3 4 5g 3.3.5 Turán Numbers for C , C , C , C . 75 f 3 4 5 6g 3.3.6 Turán Numbers for C , C , C , C , C . 75 f 3 4 5 6 7g 3.3.7 Extremal Graphs for Single Cycles, ex(n, C ) . 76 f ig 3.3.7.1 Even Cycles; ex(n, C ) . 76 f 2kg 3.3.7.2 Odd Cycles; ex(n, C ) . 77 f 2k+1g 3.3.8 Extremal Graphs with Bounded Girth . 77 3.3.9 Extremal Bipartite Graphs . 78 3.3.9.1 ex(n, C ) ........................ 78 f 4g [ B 3.3.10 Extremal Bipartite Graphs with a Bounded Girth . 79 3.4 The Generation Algorithm . 80 3.4.1 Extensions and Reductions . 80 3.4.2 Genuine Reductions . 81 Contents xiii 3.4.3 Pseudo-Code . 85 3.5 Formal Proofs . 91 3.6 Parallelisation . 96 3.7 Efficiency and Pruning the Generation Tree . 102 3.7.1 Determining the Empty Classes . 102 3.7.2 Parental Rules . 105 3.7.2.1 Graphs of Type A with Parents of Type A . 107 3.7.2.2 Number of Edges Between Vertices in mins(G) . 110 3.7.2.3 Bipartite Graphs with no C4 . 111 3.7.2.4 Bipartite Subgraphs . 113 3.7.3 Grand Parental Rules . 116 3.8 Efficiency and Group Calculations . 120 3.9 Conclusion and Future Work . 126 4 A Hierarchical Canonical labelling and its Application in Generation of Graphs 131 4.1 Introduction . 132 4.2 Hierarchical Canonical Labellings for Graphs . 136 4.2.1 The Recursive Hierarchical Canonical Labellings . 138 4.2.2 Formal Proof .