Vertex-regular $1$-factorizations in infinite graphs

TL;DR

This paper characterizes when infinite complete equipartite graphs admit vertex-regular 1-factorizations with automorphism groups, providing necessary and sufficient conditions and constructions for such factorizations.

Contribution

It establishes a complete characterization of vertex-regular 1-factorizations in infinite graphs and offers new constructions under specified group conditions.

Findings

A vertex-regular 1-factorization exists iff the automorphism group has a subgroup of order n with index m.

Provides sufficient conditions for infinite Cayley graphs to have regular 1-factorizations.

Constructs 1-factorizations containing a given subfactorization with vertex-regular automorphism groups.

Abstract

The existence of $1$-factorizations of an infinite complete equipartite graph $K_m[n]$ (with $m$ parts of size $n$) admitting a vertex-regular automorphism group $G$ is known only when $n=1$ and $m$ is countable (that is, for countable complete graphs) and, in addition, $G$ is a finitely generated abelian group $G$ of order $m$. In this paper, we show that a vertex-regular $1$-factorization of $K_m[n]$ under the group $G$ exists if and only if $G$ has a subgroup $H$ of order $n$ whose index in $G$ is $m$. 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.