On the Enumeration of One-Factorizations of Complete Graphs Containing Prescribed Automorphism Groups

Abstract

In this paper we use orderly algorithms to enumerate (perfect) one-factorizations of complete graphs, the automorphism groups of which contain certain prescribed subgroups. We showed that, for the complete graph ${K_{12}}$, excluding those one-factorizations containing exactly one automorphism of six disjoint cycles of length two, there are precisely 56391 nonisomorphic one-factorizations of ${K_{12}}$ with nontrivial automorphism groups. We also determined that there are precisely 21 perfect one-factorizations of ${K_{14}}$ that have nontrivial automorphism groups.