Edge-Transitive Homogeneous Factorisations of Complete Graphs

Abstract

This thesis concerns the study of homogeneous factorisations of complete graphs with edge-transitive factors. A factorisation of a complete graph $K_n$ is a partition of its edges into disjoint classes. Each class of edges in a factorisation of $K_n$ corresponds to a spanning subgraph called a factor. If all the factors are isomorphic to one another, then a factorisation of $K_n$ is called an isomorphic factorisation. A homogeneous factorisation of a complete graph is an isomorphic factorisation where there exists a group $G$ which permutes the factors transitively, and a normal subgroup $M$ of $G$ such that each factor is $M$-vertex-transitive. If $M$ also acts edge-transitively on each factor, then a homogeneous factorisation of $K_n$ is called an edge-transitive homogeneous factorisation. The aim of this thesis is to study edge-transitive homogeneous factorisations of $K_n$. We achieve a nearly complete explicit classification except for the case where $G$ is an affine 2-homogeneous group of the form $Z_p^R \rtimes G_0$, where $G_0 \leq ΓL(1,p^R)$. In this case, we obtain necessary and sufficient arithmetic conditions on certain parameters for such factorisations to exist, and give a generic construction that specifies the homogeneous factorisation completely, given that the conditions on the parameters hold. Moreover, we give two constructions of infinite families of examples where we specify the parameters explicitly. In the second infinite family, the arc-transitive factors are generalisations of certain arc-transitive, self-complementary graphs constructed by Peisert in 2001.