Excessive factorizations of bipartite multigraphs

Let G be a multigraph. We say that G is 1-extendable if every edge of G is contained in a 1-factor. Suppose G is 1-extendable. An excessive factorization of G is a set F = {F1, F2, . . . , Fr} of 1-factors of G whose union is E(G) and, subject to this condition, r is minimum. The integer r is called the excessive index of G and denoted by χ′e(G). Analogously, let m be a positive integer. We say that a multigraph G is [m]-coverable if every edge belongs to a matching of size m of G. An excessive [m]factorization is a set M = {M1, M2, . . . , Mk} of matchings of G, all of size m, whose union is E(G) and, subject to this condition, k is minimum. The integer k is denoted by χ′[m](G) and called excessive [m]-index of G. In this paper we shall prove that, for bipartite multigraphs, both the parameter χ′e and χ ′ [m] are computable in polynomial time, and we shall obtain an efficient algorithm to find an excessive factorization (excessive [m]-factorization, respectively) of any bipartite multigraph.