Decomposition of Complete Bipartite Digraphs and Complete Digraphs into Directed Paths and Directed Cycles of Fixed Even Length

In this paper, we give some necessary and sufficient conditions for decomposing the complete bipartite digraphs and complete digraphs into directed paths and directed cycles with arcs each. In particular, we prove that: (1) For any nonnegative integers and ; and any positive integers , , and with and ; a decomposition of into copies of and copies of exists if and only if , , , and is even when . (2) For any nonnegative integers and and any positive integers and with even and , a decomposition of into copies of and copies of exists if and only if and . We also give necessary and sufficient conditions for such decompositions to exist when or .