Transitive decompositions of graphs

A decomposition of a graph is a partition of the edge set. One can also look at partitions of the arc set but in this talk we restrict our attention to edges. If each part of the decomposition is a spanning subgraph then we call the decomposition a factorisation and the parts are called factors. Decompositions are especially interesting when the subgraphs induced by each part are pairwise isomorphic. Such decompositions are known as isomorphic decompositions. Decompositions of graphs have been widely studied and much attention has been paid to determining when a given graph can be decomposed into copies of a certain subgraph, for example, cycles or 1-factors. See for example [2, 8, 9, 10, 16]. A special class of decompositions is transitive decompositions. A G-transitive decomposition of a graph Γ is a decomposition which is invariant under some group G of automorphisms of Γ such that G acts transitively on the set of parts of the decomposition. This class of decompositions has been widely studied in many different guises. A partial linear space is a set of points and a set of subsets of the point set called lines, such that each pair of points is contained in at most one line. Given a decomposition of a graph into complete subgraphs, we can form a partial linear space with point set the set of vertices and line set the set of parts of the decomposition. Since each edge lies in only one part, every pair of points lies in at most one line, and so we indeed have a partial linear space. If the decomposition is G-transitive, then the partial linear space is G-line-transitive, and if G is also transitive on the set of arcs of the graph then the partial line space is G-flag-transitive. Conversely, given a G-line-transitive partial linear space we can construct a G-transitive decomposition of the collinearity graph of the partial linear space, that is, of the graph with vertices the set of points such that two points are adjacent if they lie on the same line. In the special case where the original graph is a complete graph then we have a linear space, that is, every two points lie on a unique line. Flag-transitive linear spaces were classified in [3]. If G is an arc-transitive group of automorphisms of the complete graph Kn then G acts 2-transitively on a set of size n. Cameron and Korchmáros [4] have determined all factorisations of complete graphs into 1-factors with a 2-transitive automorphism group, while Sibley [15] has extended this to a classification of all G-transitive decompositions of complete graphs such that G is 2-transitive. Sibley calls such decompositions 2transitive edge-coloured graphs. Recently all G-transitive decompositions of graphs with G rank 3 of product action [1] have been characterised.