Decomposition of bipartite graphs under degree constraints

Abstract

AbstractLet G = (A, B; E) be a bipartite graph. Let e1, e2 be nonnegative integers, and f1, f2 nonnegative integer‐valued functions on V(G) such that ei ≦ |E| ≦ e1 + e2 and fi(v) ≦ d(v) ≦ f1(v) + f2(v) for all v ϵ V(G) (i = 1, 2). Necessary and sufficient conditions are obtained for G to admit a decomposition in spanning subgraphs G1 = (A, B; E1) and G2 = (A, B; E2) such that |Ei| ≦ ei and dGi(v) ≦ fi(v) for all v ϵ V(G) (i = 1, 2). The result generalizes a known characterization of bipartite graphs with a k‐factor. Its proof uses flow theory and is a refinement of the proof of an analogous result due to Folkman and Fulkerson. By applying corresponding flow algorithms, the described decomposition can be found in polynomial time if it exists. As an application, an assignment problem is solved. © 1993 by John Wiley & Sons, Inc.