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

Henning and Yeo (2012) conjectured that a 3-regular digraph DD contains two vertex disjoint directed cycles of different lengths if either DD is of sufficiently large order or DD 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,v0C=v0,v1,…,vn−1,v0 and a spanning 1-circular subdigraph D(n,S)D(n,S), where S={s}S={s} with s>1s>1 and the orderings of the vertices in D(n,S)D(n,S) and in the Hamilton cycle CC are the same, does indeed have two vertex disjoint directed cycles of different lengths.