Some problems on factorizations with constraints in bipartite graphs Original Research Article

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 defined on V(G) such that g(x)⩽f(x) for each x∈V(G). A (g,f)-factor of G is a spanning subgraph F of G such that g(x)⩽dF(x)⩽f(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 k⩽g(x) for any x∈V(G) and has a (g,f)-factorization k-orthogonal to any given subgraph with km edges if k−1⩽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 0⩽g(x)⩽f(x) for any x∈V(G). Furthermore, it is shown that there are polynomial algorithms for finding the desired factorizations and the results in this paper are in some sense best possible.