Quaternionic $1-$factorizations and complete sets of rainbow spanning trees

TL;DR

This paper investigates the existence of symmetric 1-factorizations and rainbow spanning trees in complete graphs, extending known results to broader classes of groups and addressing open cases for even n.

Contribution

It generalizes previous findings by proving the existence of G-regular 1-factorizations and rainbow spanning trees for additional group classes beyond cyclic and dihedral groups.

Findings

Existence of G-regular 1-factorizations for new group classes.

Rainbow spanning trees exist under broader symmetry conditions.

Results cover cases where n is even and previously unresolved.

Abstract

A $1-$factorization of a complete graph on $2n$ vertices is said to be $G-$regular if it posseses an automorphism group $G$ acting sharply transitively on the vertex-set. The problem of determining which groups can realize such a situation dates back to a result by Hartman and Rosa (1985) on cyclic groups and, when $n$ is even, the problem is still open. An attempt to obtain a fairly precise description of groups and $1-$factorizations satisfying this symmetry constrain can be done by imposing further conditions. It was recently proved, see Rinaldi (2021) and Mazzuoccolo et al. (2019), that a $G-$regular $1-$factorization together with a complete set of rainbow spanning trees exists whenever $n$ is odd, while the existence for each $n$ even was proved when either $G$ is cyclic and $n$ is not a power of $2$, or when $G$ is a dihedral group. In this paper we extend this result and prove the existence also for other classes of groups.