Strong difference families over arbitrary graphs

The concept of a strong difference family formally introduced in Buratti [J Combin Designs 7 (1999), 406-425] with the aim of getting group divisible designs with an automorphism group acting regularly on the points, is here extended for getting, more generally, sharply-vertex-transitive Γ-decompositions of a complete multipartite graph for several kinds of graphs Γ. We show, for instance, that if Γ has e edges, then it is often possible to get a sharply-vertex-transitive Γ-decomposition of $K_{m×e}$ for any integer m whose prime factors are not smaller than the chromatic number of Γ. This is proved to be true whenever Γ admits an α-labeling and, also, when Γ is an odd cycle or the Petersen graph or the prism $T_5$ or the wheel $W_6$. We also show that sometimes strong difference families lead to regular Γ-decomposition of a complete graph. We construct, for instance, a regular cube-decomposition of $K_{16m}$ for any odd integer m whose prime factors are all congruent to 1 modulo 6.