Edge-Colorings of Weighted Graphs

Abstract

Let G be a graph with a positive integer weight ω(v) for each vertex v. One wishes to assign each edge e of G a positive integer f(e) as a color so that ω(v) ≤ |f(e) − f(e^′)| for any vertex v and any two edges e and e^′ incident to v. Such an assignment f is called an ω-edge-coloring of G, and the maximum integer assigned to edges is called the span of f. The ω-chromatic index of G is the minimum span over all ω-edge-colorings of G. In the paper, we present various upper and lower bounds on the ω-chromatic index, and obtain three efficient algorithms to find an ω-edge-coloring of a given graph. One of them finds an ω-edge-coloring with span smaller than twice the ω-chromatic index.