Some Problems on Factorizations with Constraints in Bipartite Graphs

Let G = (X; Y; E(G)) be a bipartite graph with vertex set V (G) = X ∪ Y and edge set E(G) and let g and f be two non-negative integer-valued functions de1ned on V (G) such that g(x)6f(x) for each x∈V (G). A (g; f)-factor of G is a spanning subgraph F of G such that g(x)6dF (x)6f(x) for each x∈V (F); a (g; f)-factorization of G is a partition of E(G) into edge-disjoint (g; f)-factors. In this paper it is proved that every bipartite (mg+m−1; mf− m + 1)-graph has (g; f)-factorizations randomly k-orthogonal to any given subgraph with km edges if k6 g(x) for any x∈V (G) and has a (g; f)-factorization k-orthogonal to any given subgraph with km edges if k − 16 g(x) for any x∈V (G) and that every bipartite (mg; mf)-graph has a (g; f)-factorization orthogonal to any given m-star if 06 g(x)6f(x) for any x∈V (G). Furthermore, it is shown that there are polynomial algorithms for 1nding the desired factorizations and the results in this paper are in some sense best possible. ? 2003 Elsevier Science B.V. All rights reserved.