An algorithm for the orientation of complete bipartite graphs

Let ^G be a graph with vertex set ^V(G) and edge set ^E(G). We consider the problem of orienting the edges of a complete bipartite graph ^K_n,n so only two different in-degrees ^a and ^b occur. An obvious necessary condition for orienting the edges of ^G so that only two in-degrees ^a and ^b occur, is that there exist positive integers ^s and ^t satisfying ^s+t=|V(G)| and ^as+bt=|V(G)|. In this paper, we show that the necessary condition is also sufficient for a complete bipartite graph ^K_n,n. Furthermore, we give the algorithms of orientations with only two in-degrees of ^K_n,n.