Dipartimento di Scienze di Base e Applicate per l'Ingegneria

A cycle decomposition of a graph G is a set of cycles whose edges partition the edge--set of G. The Oberwolfach problem OP(F) (with F denoting any graph on v vertices that is the vertex-disjoint union of cycles) asks for a decomposition of K_v or K_v - I (i.e., the complete graph minus a 1-factor) whose cycles can be partitioned into classes each isomorphic to F. A successful approach to tackle this problem is to require that our solution has an automorphism group G fixing one vertex and acting sharply transitively on the others, namely, it is 1-rotational under G. In the case where F consists of a single cycle, a solution to OP(F) is more commonly called a Hamiltonian cycle system of order v (briefly, HCS(v)). Although the literature on this topic is quite extensive, very little is known on the existence of an HCS(v) with a nice automorphism group. In this talk, I will present some recent results concerning these problems.