Vertex‐regular 1‐factorizations in infinite graphs. Issue 5 (8th February 2022) - Costa, Simone; Traetta, Tommaso
Abstract: The existence of 1‐factorizations of an infinite complete equipartite graph K m [ n ] K_m[n] (with m m parts of size n n ) admitting a vertex‐regular automorphism group G G is known only when n = 1 n=1 and m m is countable (i.e., for countable complete graphs) and, in addition, G G is a finitely generated abelian group G G of order m m . In this paper, we show that a vertex‐regular 1‐factorization of K m [ n ] K_m[n] under the group G G exists if and only if G G has a subgroup H H of order n n whose index in G G is m m . 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.