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 K-2n in such a way that the union of any two (cyclically) consecutive one-factors is always isomorphic to 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 greater than or equal to 1 that there is a sequentially perfect one-factorization of K-2n. As well, for any odd integer m greater than or equal to 1, we prove that there is a sequentially uniform one-factorization of K-2tm of type (4, 4,..., 4) for all integers t greater than or equal to 2 + inverted right perpendicularlog(2) minverted left perpendicular (where type (4, 4,..., 4) denotes a two-regular graph consisting of disjoint cycles of length four).