On vertex disjoint cycles of different lengths in 3-regular digraphs

Abstract

Henning and Yeo (2012) conjectured that a 3-regular digraph D contains two vertex disjoint directed cycles of different lengths if either D is of sufficiently large order or D is bipartite. In this paper, we disprove the first conjecture. Further, we give support for the second conjecture by proving that every bipartite 3-regular digraph, which either possesses a cycle factor with at least two directed cycles or has a Hamilton cycle C=v0,v1,…,vn−1,v0 and a spanning 1-circular subdigraph D(n,S), where S={s} with s>1 and the orderings of the vertices in D(n,S) and in the Hamilton cycle C are the same, does indeed have two vertex disjoint directed cycles of different lengths.