Simple Reduction of f-Colorings to Edge-Colorings

Abs t r ac t . In an edge-coloring of a graph G = (V, E) each color appears around each vertex at most once. An f-coloring is a generalization of an edge-coloring in which each color appears around each vertex v at most f(v) times where f is a function assigning a natural number f(v) e N to each vertex v E V. In this paper we first give a simple reduction of the f-coloring problem to the ordinary edge-coloring problem, that is, we show that, given a graph G = (V,E) and a function f : V -4 N, one can directly construct in polynomial-time a new simple graph whose edge-coloring using a minimum number of colors immediately induces an f-coloring of G using a minimum number of colors. As by-products, we give a necessary and sufficient condition for a graph to have an f factorization, and show that the edge-coloring problem for multigraphs can be easily reduced to edge-coloring problems for simple graphs.