Sharply transitive 1-factorizations of complete multipartite graphs

Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic

Sharply transitive 1-factorizations of complete multipartite graphs

Given a finite group G of even order, which graphs T have a 1-factorization admitting G as an automorphism group with a sharply transitive action on the vertex-set? Starting from this question we prove some general results and develop an exhustive analysis when T is a complete multipartite graph and G is cyclic