On the Directed Hamilton-Waterloo Problem with Two Cycle Sizes

TL;DR

This paper investigates the Directed Hamilton-Waterloo Problem, establishing necessary conditions and providing solutions for specific cycle size pairs in directed 2-factorizations of complete symmetric digraphs.

Contribution

It offers necessary conditions and explicit solutions for the problem with particular cycle size pairs, advancing understanding of directed 2-factorizations.

Findings

Necessary conditions for solutions are established.

Solutions are provided for specific cycle pairs.

The problem is fully solved for the listed cycle size pairs.

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 of this problem are given, and the problem is solved for the factors with $(m, n)\in \{(4,6),(4,8),(4,12),(4,16),(6,12)$,$(8,16),(3,5),(3,15),(5,15)\}$.