Given a finite group G of even order, which graphs Γ have a 1−factorization admitting G as automorphism group with a sharply transitive action on the vertex-set? Starting from this question, we prove some general results and develop an exhaustive analysis when Γ is a complete multipartite graph and G is cyclic.A 1−factor of a graph is a collection of edges such that each vertex is incident with exactly one edge. a factorization of a regular graph is a partition of the edge set of the graph into disjoint factors. If the graph has valency v, then a factorization is equivalent to a coloring of the edges in v colors (one color for each factor)...