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 DK_m,n and complete digraphs DK_n into directed paths P→k+1 and directed cycles C→k with k arcs each. In particular, we prove that: (1) For any nonnegative integers p and q; and any positive integers m, n, and k with m≥k and n≥k; a decomposition of DK_m,n into p copies of P→k+1 and q copies of (Formula presented.) exists if and only if k(p+q)=2mn, p≠1, (m,n,k,p)≠(2,2,2,3), and k is even when q>0. (2) For any nonnegative integers p and q and any positive integers n and k with k even and(Formula presented.), a decomposition of (Formula presented.) and q copies of (Formula presented.) exists if and only if k(p+q)=n(n-1) and p≠1. We also give necessary and sufficient conditions for such decompositions to exist when k=2 or 4.