An automorphism group G of a 1-factorization of the complete multipartite graph $K_{ m\times n}$ consists in permutations of the vertices of the graph mapping factors to factors. In this paper, we give a complete answer to the existence or non-existence problem of a 1-factorization of $K_{ m\times n}$ admitting an abelian group acting sharply transitively on the vertices of the graph.
Categories: math.CO