Primitive decompositions of Johnson graphs

A transitive decomposition of a graph is a partition of the edge set together with a group of automorphisms which transitively permutes the parts. In this paper we determine all transitive decompositions of the Johnson graphs such that the group preserving the partition is arc-transitive and acts primitively on the parts.Comment: 35 page

Hamiltonian cycles and 1-factors in 5-regular graphs

It is proven that for any integer $g \ge 0$ and $k \in \{ 0, \ldots, 10 \}$, there exist infinitely many 5-regular graphs of genus $g$ containing a 1-factorisation with exactly $k$ pairs of 1-factors that are perfect, i.e. form a hamiltonian cycle. For $g = 0$, this settles a problem of Kotzig from 1964. Motivated by Kotzig and Labelle's "marriage" operation, we discuss two gluing techniques aimed at producing graphs of high cyclic edge-connectivity. We prove that there exist infinitely many planar 5-connected 5-regular graphs in which every 1-factorisation has zero perfect pairs. On the other hand, by the Four Colour Theorem and a result of Brinkmann and the first author, every planar 4-connected 5-regular graph satisfying a condition on its hamiltonian cycles has a linear number of 1-factorisations each containing at least one perfect pair. We also prove that every planar 5-connected 5-regular graph satisfying a stronger condition contains a 1-factorisation with at most nine perfect pairs, whence, every such graph admitting a 1-factorisation with ten perfect pairs has at least two edge-Kempe equivalence classes. The paper concludes with further results on edge-Kempe equivalence classes in planar 5-regular graphs.Comment: 27 pages, 13 figures; corrected figure

Parity of Sets of Mutually Orthogonal Latin Squares

Every Latin square has three attributes that can be even or odd, but any two of these attributes determines the third. Hence the parity of a Latin square has an information content of 2 bits. We extend the definition of parity from Latin squares to sets of mutually orthogonal Latin squares (MOLS) and the corresponding orthogonal arrays (OA). Suppose the parity of an $\mathrm{OA}(k,n)$ has an information content of $\dim(k,n)$ bits. We show that $\dim(k,n) \leq {k \choose 2}-1$. For the case corresponding to projective planes we prove a tighter bound, namely $\dim(n+1,n) \leq {n \choose 2}$ when $n$ is odd and $\dim(n+1,n) \leq {n \choose 2}-1$ when $n$ is even. Using the existence of MOLS with subMOLS, we prove that if $\dim(k,n)={k \choose 2}-1$ then $\dim(k,N) = {k \choose 2}-1$ for all sufficiently large $N$. Let the ensemble of an $\mathrm{OA}$ be the set of Latin squares derived by interpreting any three columns of the OA as a Latin square. We demonstrate many restrictions on the number of Latin squares of each parity that the ensemble of an $\mathrm{OA}(k,n)$ can contain. These restrictions depend on $n\mod4$ and give some insight as to why it is harder to build projective planes of order $n \not= 2\mod4$ than for $n \not= 2\mod4$. For example, we prove that when $n \not= 2\mod 4$ it is impossible to build an $\mathrm{OA}(n+1,n)$ for which all Latin squares in the ensemble are isotopic (equivalent to each other up to permutation of the rows, columns and symbols)

Moduli Webs and Superpotentials for Five-Branes

We investigate the one-parameter Calabi-Yau models and identify families of D5-branes which are associated to lines embedded in these manifolds. The moduli spaces are given by sets of Riemann curves, which form a web whose intersection points are described by permutation branes. We arrive at a geometric interpretation for bulk-boundary correlators as holomorphic differentials on the moduli space and use this to compute effective open-closed superpotentials to all orders in the open string couplings. The fixed points of D5-brane moduli under bulk deformations are determined.Comment: 41 pages, 1 figur

Generating Uniformly-Distributed Random Generalised 2-designs with Block Size 3

PhDGeneralised t-designs, defined by Cameron, describe a generalisation of many combinatorial objects including: Latin squares, 1-factorisations of K2n (the complete graph on 2n vertices), and classical t-designs. This new relationship raises the question of how their respective theory would fare in a more general setting. In 1991, Jacobson and Matthews published an algorithm for generating uniformly distributed random Latin squares and Cameron conjectures that this work extends to other generalised 2-designs with block size 3. In this thesis, we divide Cameron’s conjecture into three parts. Firstly, for constants RC, RS and CS, we study a generalisation of Latin squares, which are (r c) grids whose cells each contain RC symbols from the set f1;2; : : : ; sg such that each symbol occurs RS times in each column and CS times in each row. We give fundamental theory about these objects, including an enumeration for small parameter values. Further, we prove that Cameron’s conjecture is true for these designs, for all admissible parameter values, which provides the first method for generating them uniformly at random. Secondly, we look at a generalisation of 1-factorisations of the complete graph. For constants NN and NC, these graphs have n vertices, each incident with NN coloured edges, such that each colour appears at each vertex NC times. We successfully show how to generate these designs uniformly at random when NC 0 (mod 2) and NN NC. Finally, we observe the difficulties that arise when trying to apply Jacobson and Matthews’ theory to the classical triple systems. Cameron’s conjecture remains open for these designs, however, there is mounting evidence which suggests an affirmative result. A function reference for DesignMC, the bespoke software that was used during this research, is provided in an appendix

On the Expansions in Spin Foam Cosmology

We discuss the expansions used in spin foam cosmology. We point out that already at the one vertex level arbitrarily complicated amplitudes contribute, and discuss the geometric asymptotics of the five simplest ones. We discuss what type of consistency conditions would be required to control the expansion. We show that the factorisation of the amplitude originally considered is best interpreted in topological terms. We then consider the next higher term in the graph expansion. We demonstrate the tension between the truncation to small graphs and going to the homogeneous sector, and conclude that it is necessary to truncate the dynamics as well.Comment: 17 pages, 4 figures, published versio