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

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.