Abstract
This paper introduces a new type of edge-coloring of multigraphs, called anfg-coloring, in which each color appears at each vertexv no more thanf(v) times and at each set of multiple edges joining verticesv andw no more thang(vw) times. The minimum number of colors needed tofg-color a multigraphG is called thefg-chromatic index ofG. Various upper bounds are given on thefg-chromatic index. One of them is a generalization of Vizing's bound for the ordinary chromatic index. Our proof is constructive, and immediately yields a polynomial-time algorithm tofg-color a given multigraph using colors no more than twice thefg-chromatic index.
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Nakano, S., Nishizeki, T. & Saito, N. On thefg-coloring of graphs. Combinatorica 10, 67–80 (1990). https://doi.org/10.1007/BF02122697
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DOI: https://doi.org/10.1007/BF02122697