Vertex‐regular 1‐factorizations in infinite graphs

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