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

TL;DR

This paper introduces a new algorithm for $( ext{Delta}+1)$-edge coloring that significantly reduces the running time from $ ilde O(m oot n)$ to $ ilde O(mn^{1/3})$, marking a major advancement in graph coloring efficiency.

Contribution

The paper presents the first polynomial-time improvement for $( ext{Delta}+1)$-edge coloring in over 40 years, breaking the $m oot n$ time barrier.

Findings

Achieved $ ilde O(mn^{1/3})$ running time for edge coloring.

First polynomial improvement in over four decades.

Significantly faster algorithm for a fundamental graph problem.

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.