On the Directed Hamilton-Waterloo Problem with Two Cycle Sizes
Abstract
The Directed Hamilton-Waterloo Problem asks for a directed $2$-factorization of the complete symmetric digraph $K_v^*$ 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 to this problem are given, and the problem is completely solved for the factors with $(m, n)\in \{(4,6),(4,8),(4,12),(4,16),(6,12),(8,16)\}$. Furthermore, the problem is solved for $(m, n)\in \{(3,5),(3,15),(5,15)\}$ when $v$ is odd with a few possible exceptions.
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