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.
We study arc-transitive homogeneous factorizations of a complete graph on 81 vertices and give an almost complete description of all possible factors that arise. In particular, we show that there exists an arc-transitive homogeneous factorization of K 81 such that the factors are isomorphic to the Hamming graph Hð9; 2Þ. This gives rise to an edge partition of K 81 into 90 copies of K 9 and it turns out that these copies of K 9 form the blocks of the exceptional nearfield affine plane of order 9.
A routing R in a graph consists of a simple path p uv from u to v for each ordered pair of distinct vertices (u, v). We will call R optimal if all the paths p uv are shortest paths and if edges of the graph occur equally often in the paths of R. In 1994, Solé gave a sufficient condition involving the automorphism group for a graph to have an optimal routing in this sense. Graphs which satisfy Solé's condition are called orbital regular graphs. It is often difficult to determine whether or not a given graph is orbital regular. In this paper, we give a necessary and sufficient condition for a Hamming graph to be orbital regular with respect to a certain natural subgroup of automorphisms.
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