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 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.
bipartite graph (g f)-factor algorithm orthogonal factorization