A class of starter induced 1-factorizations

Abstract

A technique of Mullin and Nemeth for constructing strong starters in certain Abelian groups of prime power order is considered. These strong starters induce 1-factorizations of complete graphs. It is easy to show that these 1-factorizations possess enough symmetry to insure that if {F_i, F_j} and {F_k, F_m} are pairs of distinct 1-factors from such a 1-factorization, then the cycle structures of F_i ∪ F_j and F_k ∪ F_m are identical. The method is applied to construct what is apparently the first example of a 1-factorization of the complete graph on 28 points, K₂₈, such that the union of every two distinct 1-factors is a Hamiltonian circuit. This problem, generalized to K_2n, has arisen in several contexts and the above result strengthens the conjecture of various authors that such 1-factorizations exist on all K_2n. The first unsettled case is now K₃₆.