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

TL;DR

This work addresses the classic problem of computing a $(\Delta+1)$-edge coloring for a simple graph with $n$ vertices and $m$ edges. It introduces a novel canonical-instance framework that isolates a star-centered residual structure, enabling the uncolored edges to be handled via short alternating-path flips and limited fan rotations. By combining a randomized edge-sampling template, an Eulerian-partition-based slack coloring step, and a careful, case-based analysis of Vizing fans, the authors achieve a randomized runtime of $\tilde{O}(mn^{1/3})$, beating the four-decade $m\sqrt{n}$ barrier. The results extend the toolkit for edge coloring, leveraging structure in near-regular subgraphs and a star-aggregation approach that may influence future algorithms for related coloring and scheduling problems. Overall, the paper delivers the first polynomial improvement for this foundational problem and demonstrates a scalable strategy that integrates probabilistic methods, graph partitioning, and Vizing-style color-extension techniques.

Abstract

Vizing's theorem states that any $n$-vertex $m$-edge graph of maximum degree $Δ$ can be {\em edge colored} using at most $Δ+ 1$ 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.

Figures (4)

• Figure 1: In this picture, A vertex $s$ is directed to $t$ if $\psi(u, t)\in \mathsf{miss}_\psi(s)$. Applying Vizing' procedure for either $(u, v)$ or $(u, v')$ would end up with the same alternating path starting with same $\{x, \psi(u, w_2)\}$-alternating path. We will show that the $\{x, \psi(u, w_1)\}$-alternating path can also be used for color extension.
• Figure 2: If the neighborhood of $u$ is a star rather than a chain, then we could virtually rotate uncolored edges to $(u, w_0), (u, w_1)$. As $w_0, w_1$ are both missing blue around them, we can pair them together and reuse the argument for the pairing the clients case.
• Figure 3: If $w$ is the target of many clients in different star subgraphs, we can show that all these star centers have the same type of alternating paths which begin with blue edges.
• Figure 8: This example shows a tree component $T$ where $D\setminus F$ are red vertices, $F$ are orange vertices, and branching vertices are blue ones. So by definition, $\mathsf{br}_\psi(T) = 4 = 1+|D\setminus F|$.

Theorems & Definitions (41)

• Theorem 1.1
• Lemma 2.1
• Lemma 3.1
• proof
• Lemma 3.2: Eulerian partition arjomandi1982efficientgabow1985algorithmssinnamon2019fast
• Lemma 3.3: elkin2024deterministic
• proof
• Theorem 4.1
• Claim 4.1
• proof