Abelian 1-factorizations of complete multipartite graphs

TL;DR

This paper investigates the conditions under which complete multipartite graphs can have 1-factorizations that admit an abelian automorphism group acting sharply transitively on vertices, providing a complete characterization.

Contribution

It offers a complete characterization of the existence of abelian 1-factorizations of complete multipartite graphs, solving an open problem in graph theory.

Findings

Characterization of when abelian 1-factorizations exist for $K_{m\times n}$

Conditions for abelian groups acting sharply transitively on vertices

Complete solutions for the existence or non-existence of such factorizations

Abstract

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.