Automorphisms generating disjoint Hamilton cycles in star graphs

In the first part of the thesis we define an automorphism φn for each star graph Stn of degree n − 1, which yields permutations of labels for the edges of Stn taken from the set of integers {1, . . . , bn/2c}. By decomposing these permutations into permutation cycles, we are able to identify edge-disjoint Hamilton cycles that are automorphic images of a known two-labelled Hamilton cycle H1 2(n) in Stn. Our main result is an improvement from the existing lower bound of bϕ(n)/10c to b2ϕ(n)/9c, where ϕ is Euler’s totient function, for the known number of edge-disjoint Hamilton cycles in Stn for all odd integers n. For prime n, the improvement is from bn/8c to bn/5c. We extend this result to the cases when n is the power of a prime other than 3 and 7. The second part of the thesis studies ‘symmetric’ collections of edge-disjoint Hamilton cycles in Stn, i.e. collections that comprise images of H1 2(n) under general label-mapping automorphisms. We show that, for all even n, there exists a symmetric collection of bϕ(n)/2c edge-disjoint Hamilton cycles, and Stn cannot have symmetric collections of greater than bϕ(n)/2c such cycles for any n. Thus, Stn is not symmetrically Hamilton decomposable if n is not prime. We also give cases of even n, in terms of Carmichael’s reduced totient function λ, for which ‘strongly’ symmetric collections of edge-disjoint Hamilton cycles, which are generated from H1 2(n) by a single automorphism, can and cannot attain the optimum bound bϕ(n)/2c for symmetric collections. In particular, we show that if n is a power of 2, then Stn has a spanning subgraph with more than half of the edges of Stn, which is strongly symmetrically Hamilton decomposable. For odd n, it remains an open problem as to whether the bϕ(n)/2c can be achieved for symmetric collections, but we are able to show that, for certain odd n, a ϕ(n)/4 bound is achievable and optimal for strongly symmetric collections. The search for edge-disjoint Hamilton cycles in star graphs is important for the design of interconnection network topologies in computer science. All our results improve on the known bounds for numbers of any kind of edge-disjoint Hamilton cycles in star graphs

A graph partition problem

Given a graph $G$ on $n$ vertices, for which $m$ is it possible to partition the edge set of the $m$-fold complete graph $mK_n$ into copies of $G$? We show that there is an integer $m_0$, which we call the \emph{partition modulus of $G$}, such that the set $M(G)$ of values of $m$ for which such a partition exists consists of all but finitely many multiples of $m_0$. Trivial divisibility conditions derived from $G$ give an integer $m_1$ which divides $m_0$; we call the quotient $m_0/m_1$ the \emph{partition index of $G$}. It seems that most graphs $G$ have partition index equal to $1$, but we give two infinite families of graphs for which this is not true. We also compute $M(G)$ for various graphs, and outline some connections between our problem and the existence of designs of various types

Uniformity in association schemes and coherent configurations: cometric Q-antipodal schemes and linked systems

Inspired by some intriguing examples, we study uniform association schemes and uniform coherent configurations, including cometric Q-antipodal association schemes. After a review of imprimitivity, we show that an imprimitive association scheme is uniform if and only if it is dismantlable, and we cast these schemes in the broader context of certain --- uniform --- coherent configurations. We also give a third characterization of uniform schemes in terms of the Krein parameters, and derive information on the primitive idempotents of such a scheme. In the second half of the paper, we apply these results to cometric association schemes. We show that each such scheme is uniform if and only if it is Q-antipodal, and derive results on the parameters of the subschemes and dismantled schemes of cometric Q-antipodal schemes. We revisit the correspondence between uniform indecomposable three-class schemes and linked systems of symmetric designs, and show that these are cometric Q-antipodal. We obtain a characterization of cometric Q-antipodal four-class schemes in terms of only a few parameters, and show that any strongly regular graph with a ("non-exceptional") strongly regular decomposition gives rise to such a scheme. Hemisystems in generalized quadrangles provide interesting examples of such decompositions. We finish with a short discussion of five-class schemes as well as a list of all feasible parameter sets for cometric Q-antipodal four-class schemes with at most six fibres and fibre size at most 2000, and describe the known examples. Most of these examples are related to groups, codes, and geometries.Comment: 42 pages, 1 figure, 1 table. Published version, minor revisions, April 201

Uniformity in Association schemes and Coherent Configurations: Cometric Q-Antipodal Schemes and Linked Systems

2010 Mathematics Subject Classification. Primary 05E30, Secondary 05B25, 05C50, 51E12

Computing the chromatic number of t-(v,k,[lambda]) designs

Colouring t-designs has previously been shown to be an NP-complete problem; heuristics and a practical algorithm for this problem were developed for this thesis; the algorithm was then employed to find the chromatic numbers of the sixteen non- isomorphic 2-(25, 4, 1) designs and the four cyclic 2-(19, 3, 1) designs. This thesis additionally examines the existing literature on colouring and finding chromatic numbers of t-designs

Benson\u27s Theorem for Partial Geometries

In 1970 Clark Benson published a theorem in the Journal of Algebra stating a congruence for generalized quadrangles. Since then this theorem has been expanded to other specific geometries. In this thesis the theorem for partial geometries is extended to develop new divisibility conditions for the existence of a partial geometry in Chapter 2. Then in Chapter 3 the theorem is applied to higher dimensional arcs resulting in parameter restrictions on geometries derived from these structures. In Chapter 4 we look at extending previous work with partial geometries with α = 2 to uncover potential partial geometries with higher values of α. Finally the theorem is extended to strongly regular graphs in Chapter 5. In addition we obtain expressions for the multiplicities of the eigenvalues of matrices related to the adjacency matrices of these graphs. Finally, a four lesson high school level enrichment unit is included to provide students at this level with an introduction to partial geometries, strongly regular graphs, and an opportunity to develop proof skills in this new context

Difference Sets and the Symmetric Difference Property

With the intent to discover block designs from groups of order 256 and higher that have the symmetric difference property, we analyze these block designs through products of groups that give smaller block designs with the symmetric difference property (SDP). This research expands upon the knowledge of SDP designs by looking at designs that come from groups of extremely high orders, which we analyze by looking at the difference sets of these groups. This will give way to new SDP designs that can be analyzed and studied in the near future