Vertex-regular 1-factorizations in infinite graphs

The existence of 1-factorizations of an infinite complete equipartite graph (Formula presented.) (with (Formula presented.) parts of size (Formula presented.)) admitting a vertex-regular automorphism group (Formula presented.) is known only when (Formula presented.) and (Formula presented.) is countable (i.e., for countable complete graphs) and, in addition, (Formula presented.) is a finitely generated abelian group (Formula presented.) of order (Formula presented.). In this paper, we show that a vertex-regular 1-factorization of (Formula presented.) under the group (Formula presented.) exists if and only if (Formula presented.) has a subgroup (Formula presented.) of order (Formula presented.) whose index in (Formula presented.) is (Formula presented.). Furthermore, we provide a sufficient condition for an infinite Cayley graph to have a regular 1-factorization. Finally, we construct 1-factorizations that contain a given subfactorization, both having a vertex-regular automorphism group