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Vizing's theorem states that any n-vertex m-edge graph of maximum degree can be edge colored using at most + 1 different colors Diskret. ~Analiz, '64. Vizing's original proof is algorithmic and shows that such an edge coloring can be found in O (mn) time. This was subsequently improved to O (mn), independently by Arjomandi 1982 and by Gabow et al. ~1985. In this paper we present an algorithm that computes such an edge coloring in O (mn^1/3) time, giving the first polynomial improvement for this fundamental problem in over 40 years.
Bhattacharya et al. (Fri,) studied this question.