The Directed Uniform Hamilton-Waterloo Problem Involving Even Cycle Sizes

Abstract

In this paper, factorizations of the complete symmetric digraph $K_v^{\ast }$ into uniform factors consisting of directed even cycle factors are studied as a generalization of the undirected Hamilton-Waterloo Problem. It is shown, with a few possible exceptions, that $K_v^{\ast }$ can be factorized into two nonisomorphic factors, where these factors are uniform factors of $K_v^{\ast }$ involving $K_2^{\ast }$ or directed $m$-cycles, and directed $m$-cycles or $2m$-cycles for even $m$.