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 de-composition 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 decomposi-tion. 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 decompositio