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 Kv & lowast; 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) is an element of {(4, 6), (4, 8), (4, 12), (4, 16), (6, 12), (8, 16)}. Furthermore, the problem is solved for (m, n) is an element of {(3, 5), (3, 15), (5, 15)} when v is odd with a few possible exceptions.