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.