Disjoint cycles with different length in 4-arc-dominated digraphs

A d-arc-dominated digraph is a digraph D of minimum out-degree d such that for every arc (x,y) of D, there exists a vertex u of D of out-degree d such that (u,x) and (u,y) are arcs of D. Henning and Yeo [Vertex disjoint cycles of different length in digraphs, SIAM J. Discrete Math. 26 (2012) 687-694] conjectured that a digraph with minimum out-degree at least four contains two vertex-disjoint cycles of different length. In this paper, we verify this conjecture for 4-arc-dominated digraphs.