A polynomial algorithm for finding (g, f)-colorings orthogonal to stars in bipartite graphs

Let G be a bipartite graph with vertex set V(G) and edge set E(G), and let g and f be two nonnegative integer-valued functions defined on V(G) such that g(x) <= f (x) for every vertex x of V(G). A (g, f)-coloring of G is a generalized edge-coloring in which each color appears at each vertex x at least g(x) and at most f (x) times. In this paper a polynomial algorithm to find a (g, f)-coloring of a bipartite graph with some constraints using the minimum number of colors is given. Furthermore, we show that the results in this paper are best possible.