On the Directed Hamilton-Waterloo Problem with Two Cycle Sizes

The Directed Hamilton-Waterloo Problem asks for a directed $2$-factorization of the complete symmetric digraph $K_v^{\ast }$ where there are two non-isomorphic $2$-factors. In the uniform version of the problem, factors consist of either directed $m$-cycles or $n$-cycles. In this paper, necessary conditions for a solution of this problem are given, and the problem is solved for the factors with $(m,n)\in \{(4,6),(4,8),(4,12),(4,16),(6,12)$,$(8,16),(3,5),(3,15),(5,15)\}$.