PROOF OF A CONJECTURE OF HENNING AND YEO ON VERTEX-DISJOINT DIRECTED CYCLES

M.A. Henning and A. Yeo conjectured in [SIAM J. Discrete Math., 26 (2012), pp. 687-694] that a digraph of minimum out-degree at least 4, contains two vertex-disjoint cycles of different lengths. In this paper we prove this conjecture. The main tool is a new result (to our knowledge) asserting that in a digraph D of minimum out-degree at least 4, there exist two vertex-disjoint cycles C-1 and C-2, a path P-1 from a vertex x of C-1 to a vertex z not in V(C-1) boolean OR V(C-2), and a path P-2 from a vertex y of C-2 to z, such that V(P-1) boolean AND (V(C-1) boolean OR V(C-2)) = {x}, V(P-2) boolean AND (V(C-1) boolean OR V(C-2)) = {y}, and V(P-1) boolean AND V(P-2) = {z}. In the last section, a conjecture will be proposed.