Sequentially Perfect and Uniform One-Factorizations of the Complete Graph

In this paper, we consider a weakening of the definitions of uniform and perfect one-factorizations of the complete graph. Basically, we want to order the 2n − 1 one-factors of a one-factorization of the complete graph K2n in such a way that the union of any two (cyclically) consecutive one-factors is always isomorphic to the same two-regular graph. This property is termed sequentially uniform; if this two-regular graph is a Hamiltonian cycle, then the property is termed sequentially perfect. We will discuss several methods for constructing sequentially uniform and sequentially perfect one-factorizations. In particular, we prove for any integer n ≥ 1 that there is a sequentially perfect one-factorization of K2n. As well, for any odd integer m ≥ 1, we prove that there is a sequentially uniform one-factorization of K2tm of type (4, 4, . . . , 4) for all integers t ≥ 2 + dlog2 me (where type (4, 4, . . . , 4) denotes a two-regular graph consisting of disjoint cycles of length four).