2021/06/17 by Simone Costa, Costa, Simone, Tommaso Traetta +1
Mathematics · #05C63 #05C70 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05C63 #msc:05C70
paper · pdf · doi:10.48550/arxiv.2106.09468
arxiv created 2021/06/17 · arxiv updated 2021/06/18
The existence of 1-factorizations of an infinite complete equipartite graph Km[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 Km[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.