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

Henning and Yeo [SIAM J. Discrete Math. 26 (2012) 687–694] conjectured that a 3-regular digraph D contains two vertex disjoint directed cycles of different length 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, does indeed have two vertex disjoint directed cycles of different length.