P3-factorization of complete bipartite symmetric digraphs

In path factorization, H. Wang [1] gives the necessary and sufficient conditions for the existence of P_k-factorization of a complete bipartite graph for k, an even integer. Further, Beiling Du [2] extended the work of H. Wang, and studied the P_2k-factorization of complete bipartite multigraph. For odd value of k the work on factorization was done by a number of researchers. P_3-factorization of complete bipartite graph was studied by K. Ushio [3]. P_5-factorization of complete bipartite graph was studied by J. Wang [4]. In the present paper, we study (P_5 ) ?-factorization of complete bipartite symmetric digraphs and show that the necessary and sufficient conditions for the existence of (P_5 ) ?-factorization of complete bipartite symmetric digraphs are: (1)3m?2n, (2) 3n?2m, (3)m+n?0(mod 5) and (4)5mn/2(m+n) is an integer. Mathematics Subject Classification 68R10, 05C70, 05C38.