Homogeneous factorisations of complete graphs with edge-transitive factors

Abstract

A factorisation of a complete graph K n is a partition of its edges with each part corresponding to a spanning subgraph (not necessarily connected), called a factor. A factorisation is called homogeneous if there are subgroups M < G ≤ S n such that M is vertex-transitive and fixes each factor setwise, and G permutes the factors transitively. We classify the homogeneous factorisations of K n for which there are such subgroups G, M with M transitive on the edges of a factor as well as the vertices. We give infinitely many new examples.

On generalised Paley graphs and their automorphism groups

“…Although this gives a lot of information about the group, it does not determine it completely. In a special case, that is relevant to our work on homogeneous factorisations in [11], we were able to show that the automorphism group of GPaley(q, q−1 k ) is indeed equal to the automorphism group of Cyc(q, k) (and no larger), see Theorem 1.2 (4).…”

“…We prove that in the connected, non-Hamming case, the automorphism group of a generalised Paley graph is a primitive group of affine type, and we find sufficient conditions under which the group is equal to the one-dimensional affine group of the associated cyclotomic association scheme. The results have been applied in [11] to distinguish between cyclotomic schemes and similar twisted versions of these schemes, in the context of homogeneous factorisations of complete graphs.…”

Homogeneous factorisations of Johnson graphs

“…In order to prove Theorem 5.1, we first make the following observation that uses the fact that M 0 G 0 . Moreover, as M 0 is transitive on the nonzero elements of GF(q), [16,Lemma 4.7] …”

“…Since G α contains t 1,1 it follows that G α = R α . Thus by [16,Lemma 4.7], G = AΓL(1, q). Let ω be a primitive element of GF(q).…”

“…Let ω be a primitive element of GF(q). Then by [16,Lemma 4.4] there exist unique integers d, e and s such that M 0 = t ω d ,0 , t ω e ,0 σ s with 0 ≤ e < d and satisfying particular divisibility conditions. Without loss of generality we may suppose that M is the kernel of the action of G on E and so M 0 is invariant under G α .…”