Perfect 1-factorisations of $K_{18}$ with a non-trivial automorphism group

Abstract

A perfect 1-factorisation (P1F) of $K_{2n}$ is a partition of its edges into $2n-1$ perfect matchings, any two of which form a Hamiltonian cycle. Exhaustive classification is known only up to $K_{16}$. We present a method for the exhaustive generation, up to isomorphism, of perfect 1-factorisations with a non-trivial automorphism group, which reduces the search tree far enough to make $K_{18}$ and beyond feasible. It combines two ideas. For automorphism groups of order greater than 2, classical results of Ihrig on the automorphisms of P1Fs are used as tools that reject almost all prescribed cycle types without search. For the order-2 case, which dominates, candidates are recalculated from inherited compatibility lists, in the spirit of Gill and Wanless, with an outer loop over two further rows that keeps those lists small, and Ihrig's results restrict the first factor. As the numerical proof of the method we compute all 10,710 P1Fs of $K_{18}$ with a non-trivial automorphism group; before, only six were known.