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arXiv · 2405.15449

Faster $(\Delta + 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier

Sayan Bhattacharya·Din Carmon·Martín Costa·Shay Solomon·Tianyi Zhang
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Abstract

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

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Sayan Bhattacharya, Din Carmon, Martín Costa, Shay Solomon, Tianyi Zhang. 2024-05-24. Faster $(\Delta + 1)$-Edge Coloring: Breaking the $m \sqrt{n}$ Time Barrier. https://arxiv.org/abs/2405.15449

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