Spanning Tree Decompositions of Complete Graphs Orthogonal to Rotational 1-Factorizations

Abstract

In Krussel et al. (ARS Comb 57:77–82, 2000), Krussel, Marshall, and Verall proved that whenever $2n-1$ is a prime of the form $8m+7$, there exists a spanning tree decomposition of $K_{2n}$ orthogonal to the 1-factorization $GK_{2n}$. In this paper, we develop a technique for constructing spanning tree decompositions that are orthogonal to rotational 1-factorizations of $K_{2n}$. We apply our results to show that, for every $n>2$, there exists a spanning tree decomposition orthogonal to $GK_{2n}$. We include similar applications to other rotational families of 1-factorizations, and provide directions for further research.