k-Cycle Free One-Factorizations of Complete Graphs

A one-factorization of a regular graph G is uniform if the union of any two one-factors is isomorphic to the same two-factor H, which is a disjoint union of even cycles. There are only several infinite classes of known uniform one-factorizations of complete graphs. An opposite property may be dealt with; one can ask about the existence of one-factorization such that the union of any two one-factors does not include cycles of given lengths. A onefactorization F = {F1, F2, . . . , Ft} of G is said k-cycle free if the union of any two one-factors does not include the cycle Ck as a component. Consequently, F is k -cycle free if the union of any two one-factors does not include all cycles of lengths ≤ k. It is proved that for every n ≥ 3 and every even k ≥ 4, where k 6= 2n, there exists a k-cycle free one-factorization of the complete graph K2n. Moreover, some infinite classes of k -cycle free one-factorizations of K2n are constructed.