Block-Intersection Graphs of Steiner Triple Systems by Sally Shaul Kazin (Under the Direction of E. Rodney Canfield) Abstract A Steiner triple system of order n is a collection of subsets of size three, taken from the n-element set f0; 1; :::; n − 1g; such that every pair is contained in exactly one of the subsets. The subsets are called triples, and a block-intersection graph is constructed by having each triple correspond to a vertex. If two triples have a non-empty intersection, an edge is inserted between their vertices. It is known that there are eighty Steiner triple systems of order 15 up to isomorphism. In this paper, we attempt to distinguish the eighty systems using their block-intersection graphs, as well as discuss general properties of block-intersection graphs of Steiner triple systems. Index words: Steiner triple system, Block-intersection graph, Isomorphism, Spectra Block-Intersection Graphs of Steiner Triple Systems by Sally Shaul Kazin B.S., The University of Georgia, 2005 A Thesis Submitted to the Graduate Faculty of The University of Georgia in Partial Fulfillment of the Requirements for the Degree Master of Arts Athens, Georgia 2012 c 2012 Sally Shaul Kazin All Rights Reserved Block-Intersection Graphs of Steiner Triple Systems by Sally Shaul Kazin Approved: Major Professor: E. Rodney Canfield Committee: Edward Azoff Leonard Chastkofsky Electronic Version Approved: Maureen Grasso Dean of the Graduate School The University of Georgia May 2012 Block-Intersection Graphs of Steiner Triple Systems Sally Shaul Kazin April 24, 2012 Acknowledgments I would like to thank Dr. Canfield for agreeing to be my advisor and for all of his time and effort in helping me research and write this thesis. I also want to thank my parents for everything, as I would not have made it here without them. Finally, thank you to MK, KSB, ZCS, and TH for helping me get through graduate school. You know who you are. iv Contents 1 Introduction 1 1.1 Combinatorial Definitions and Background . 2 1.2 Graph Theory Definitions . 3 1.3 Block-Intersection Graphs . 4 1.4 Properties of the Block-Intersection Graphs . 5 2 Methods of Construction 10 2.1 The Doubling Construction . 10 2.2 Lexicographically Least STS(n) . 11 2.3 Other Methods . 14 3 Cliques 17 3.1 Maximal Cliques in a BIG . 18 3.2 Maximum Cliques for n =15........................... 19 3.3 Clique Census . 21 3.4 System 1 for n =15................................ 23 4 Independent Sets 26 4.1 Independent Sets in a BIG . 26 4.2 Independent Set Census . 30 v 5 Spectra 33 5.1 Definitions and Background . 33 5.2 Spectra of the BIGs . 34 5.3 Graph Complements and Line Graphs . 35 5.4 Eigenspaces . 37 5.5 Distance-Regularity . 39 6 Differentiating Systems 41 6.1 Block-Intersection Graphs . 41 6.2 Cycle Structure . 43 6.3 Non-Graphical Methods . 44 6.4 Summary . 49 Appendix A Steiner Triple Systems of Order 13 50 Appendix B Code Used to Generate Block-Intersection Graphs 51 Bibliography 53 vi List of Tables 3.1 Clique Census for n =13............................. 21 3.2 Partial Clique Census for n = 15 . 22 4.1 Maximum Independent Sets in the BIG of STS(15) #61 . 29 4.2 Independent Set Census for n = 13 . 30 5.1 Spectra of the Block-Intersection Graphs . 35 6.1 Systems Differentiated Using Both Censuses . 42 6.2 Cycle Vectors for Select STS(15)s . 44 vii List of Figures 1.1 The BIG for STS(9) . 4 3.1 PG(3,2) . 24 3.2 The Fano Plane . 25 viii Chapter 1 Introduction In 1847, Reverend Thomas P. Kirkman published \On a Problem in Combinations" [11] that first answered questions on the existence of what would later be called Steiner triple systems. The problem he considered was to find a collection of subsets of size three, taken from the n-element set f0; 1; :::; n − 1g, with the property that every pair is contained in exactly one of the subsets. This problem was a specific case of a more general question on combinations posed by W.S.B. Woodhouse in 1844. Several years later, Jakob Steiner, apparently unaware of Kirkman's article, similarly asked about the existence of various designs, including the one solved by Kirkman. It was Steiner's name that was eventually used for the systems, although Steiner himself did not offer any solutions. Such systems were an early example of what are now known as balanced incomplete block designs in the field of combinatorial design. Kirkman's name did become associated with a related topic that originated with \Kirk- man's schoolgirl problem," in which he wanted to arrange fifteen girls into sets of three such that each girl was paired with every other girl once. This describes a Steiner triple system, but Kirkman added the condition that the thirty-five sets of three be expressible as seven partitions of f0; 1; :::; 14g. The triple system would then describe a week-long schedule for 1 daily walks where the fifteen girls were in different sets each day. In the same year his question was published, Kirkman offered a solution, as did Arthur Cayley [5]. Kirkman also asked about orders other than 15 for which an analogous problem could be solved. These designs are a subtype of Steiner triple systems often called Kirkman triple systems, and we will revisit this idea below. 1.1 Combinatorial Definitions and Background A Steiner triple system of order n is a pair (V; B) comprising the n-set V = f0; 1; :::; n − 1g and a set B of sets of size three, each a subset of V , such that any pair of elements of V appears in exactly one set of B. The sets in B are called triples or blocks. We will abbreviate \Steiner triple system of order n" as STS(n). A necessary and sufficient condition for the existence of a Steiner triple system of order n was proved by Kirkman in his 1847 paper, that n ≡ 1; 3 mod 6 [11]. As a balanced incomplete block design, an STS(n) is a (n; 3; 1)-design, where 3 is the size of a block and 1 is the number of blocks in which a pair of elements appears. In any n(n−1) n 3 STS(n), there are 6 triples. We obtain this total by computing 2 / 2 , as there are n 3 2 ways to select unordered pairs from the n-set and each triple contains 2 pairs. The replication number of an STS(n) is defined as the number r of triples in which each element of V appears. A theorem on block designs states that (n − 1) = r(3 − 1) for an STS(n), n−1 yielding the replication number equal to 2 . The subtype of STS(n)s known as Kirkman triple systems, as described above with Kirkman's schoolgirl problem, have the additional property of resolvability. First, we define a parallel class in an STS(n) as a set of pairwise disjoint triples that partitions the n-set; the union of the triples in a parallel class will contain each element once. An STS(n) is resolvable if the triples of B can be partitioned into parallel classes. In Kirkman's problem, each day 2 represents a parallel class. A necessary and sufficient condition on n for the existence of a resolvable STS(n) was not proved until 1965 [5], but it is quite simple: n ≡ 3 mod 6. Up to isomorphism, there is one STS each for orders 3, 7, and 9. There are two STS(13)s, eighty STS(15)s, over 11 billion STS(19)s, and the numbers grow exponentially from there (and are not completely determined for orders over 19) [10]. Richard Wilson showed that the number N of Steiner triple systems on a permissible order n is such that (e−5n)n2=6 ≤ N ≤ (e−1n)n2=6 [18]. In this paper we will focus on the Steiner triple systems of order 13 and 15, in particular on distinguishing among isomorphism types. The STS(15)s have been known for over half a century and have a standard numbering in the literature, which can be found in [5]. In addition, the two STS(13)s are included in Appendix A in the order by which they will be referenced. 1.2 Graph Theory Definitions A graph is an object made up of vertices and edges, where edges are links between vertices. If two vertices have an edge between them, they are adjacent. The number of edges leaving a vertex is called the degree of that vertex, and a simple graph contains no loops and at most one edge per pair of vertices; all graphs considered here will be simple. A graph with every vertex having equal degree is called regular. If for each pair of vertices, there is a sequence of edges starting from one vertex and ending at the other, the graph is connected. The complement G of a graph G is formed using the same vertex set as G with edges inserted between vertices that were not adjacent in G. For the following definitions, consider a graph on n vertices. If every pair of vertices is joined by an edge, the graph is complete and denoted by Kn. A graph can also be represented in the form of an adjacency matrix, which is an n × n matrix with entry aij equal to the 3 number of edges between the ith and jth vertices. In a simple graph, this matrix is composed entirely of 0's and 1's.