Sufficient conditions for a balanced bipartite digraph to be even pancyclic.

Let D be a strongly connected balanced bipartite directed graph of order 2 a ≥ 8 . In this note we prove: (i). If D contains a cycle of length 2 a − 2 ≥ 6 and m a x { d ( x ) , d ( y ) } ≥ 2 a − 2 for every pair of vertices { x , y } with a common out-neighbour, then for every k , 1 ≤ k ≤ a − 1 , D contains a cycle of length 2 k . (ii). If D is not a directed cycle and m a x { d ( x ) , d ( y ) } ≥ 2 a − 1 for every pair of vertices { x , y } with a common out-neighbour, then for every k , 1 ≤ k ≤ a , D contains a cycle of length 2 k unless D is isomorphic to a certain digraph of order eight which we specify. [ABSTRACT FROM AUTHOR]