Even Faster $(\Delta + 1)$-Edge Coloring via Shorter Multi-Step Vizing Chains

TL;DR

This paper introduces a faster algorithm for $( ext{Delta}+1)$-edge coloring of graphs, improving the time complexity by developing shorter multi-step Vizing chains for edge recoloring.

Contribution

The paper presents a novel algorithm achieving $ ilde O(mn^{1/4})$ time for edge coloring, with a new subroutine that constructs shorter multi-step Vizing chains.

Findings

Achieves $ ilde O(mn^{1/4})$ time complexity for edge coloring.

Introduces a subroutine for faster edge extension within $ ilde O( ext{Delta}^2 + oot{ ext{sqrt}}{ ext{Delta} n})$ time.

Produces significantly shorter multi-step Vizing chains for large $ ext{Delta}$.

Abstract

Vizing's Theorem from 1964 states that any $n$-vertex $m$-edge graph with maximum degree $\Delta$ can be {\em edge colored} using at most $\Delta + 1$ colors. For over 40 years, the state-of-the-art running time for computing such a coloring, obtained independently by Arjomandi [1982] and by Gabow, Nishizeki, Kariv, Leven and Terada~[1985], was $\tilde O(m\sqrt{n})$. Very recently, this time bound was improved in two independent works, by Bhattacharya, Carmon, Costa, Solomon and Zhang to $\tilde O(mn^{1/3})$, and by Assadi to $\tilde O(n^2)$. In this paper we present an algorithm that computes such a coloring in $\tilde O(mn^{1/4})$ time. Our key technical contribution is a subroutine for extending the coloring to one more edge within time $\tilde O(\Delta^2 + \sqrt{\Delta n})$. The best previous time bound of any color extension subroutine is either the trivial $O(n)$, dominated by the length of a Vizing chain, or the bound $\tilde{O}(\Delta^6)$ by Bernshteyn [2022], dominated by the length of {\em multi-step Vizing chains}, which is basically a concatenation of multiple (carefully chosen) Vizing chains. Our color extension subroutine produces significantly shorter multi-step Vizing chains than in previous works, for sufficiently large $\Delta$.