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A Simple Algorithm for Near-Vizing Edge-Coloring in Near-Linear Time

Abhishek Dhawan

TL;DR

The paper presents a simple randomized sequential algorithm for near-Vizing edge-coloring that achieves a $(1+\varepsilon)\Delta$-edge-coloring in near-linear time for dense graphs with $\Delta=\Omega(\log n/\varepsilon)$. It uses a two-stage strategy: Stage 1 constructs a partial coloring by sampling small palettes and applying short Vizing chains, while Stage 2 completes the coloring via a folklore $(2+\varepsilon)\Delta$-edge-coloring method, with a flagging mechanism to bound uncolored edges. The key contribution is a clean, implementable approach that yields a running time of $$O(m\log^3 n/\varepsilon^3)$$ with high probability, along with a probabilistic analysis showing that the uncolored subgraph after Stage 1 has maximum degree $O(\varepsilon\Delta)$. The work emphasizes simplicity over fastest-known runtimes, and it lays out potential avenues for extending the technique to multigraphs or achieving tighter dependence on $\varepsilon$. Overall, the paper offers a practical, theory-backed route to near-Vizing edge-coloring in dense graphs and highlights opportunities for further refinement.

Abstract

We present a simple $(1+\varepsilon)Δ$-edge-coloring algorithm for graphs of maximum degree $Δ= Ω(\log n / \varepsilon)$ with running time $O\left(m\,\log^3 n/\varepsilon^3\right)$. Our algorithm improves upon that of [Duan, He, and Zhang; SODA19], which was the first near-linear time algorithm for this problem. While our results are weaker than the current state-of-the-art, our approach is significantly simpler, both in terms of analysis as well as implementation, and may be of practical interest.

A Simple Algorithm for Near-Vizing Edge-Coloring in Near-Linear Time

TL;DR

The paper presents a simple randomized sequential algorithm for near-Vizing edge-coloring that achieves a -edge-coloring in near-linear time for dense graphs with . It uses a two-stage strategy: Stage 1 constructs a partial coloring by sampling small palettes and applying short Vizing chains, while Stage 2 completes the coloring via a folklore -edge-coloring method, with a flagging mechanism to bound uncolored edges. The key contribution is a clean, implementable approach that yields a running time of with high probability, along with a probabilistic analysis showing that the uncolored subgraph after Stage 1 has maximum degree . The work emphasizes simplicity over fastest-known runtimes, and it lays out potential avenues for extending the technique to multigraphs or achieving tighter dependence on . Overall, the paper offers a practical, theory-backed route to near-Vizing edge-coloring in dense graphs and highlights opportunities for further refinement.

Abstract

We present a simple -edge-coloring algorithm for graphs of maximum degree with running time . Our algorithm improves upon that of [Duan, He, and Zhang; SODA19], which was the first near-linear time algorithm for this problem. While our results are weaker than the current state-of-the-art, our approach is significantly simpler, both in terms of analysis as well as implementation, and may be of practical interest.
Paper Structure (14 sections, 10 theorems, 27 equations, 6 figures, 1 table, 2 algorithms)

This paper contains 14 sections, 10 theorems, 27 equations, 6 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Background and Results
  3. Discussion of Prior Work
  4. Informal Overview
  5. Concluding Remarks
  6. Notation and Preliminaries
  7. Vizing Chains
  8. Data Structures
  9. A Folklore $(2+\varepsilon)\Delta$-Edge-Coloring Algorithm
  10. Proof of Theorem \ref{['theo:main_theo']}
  11. Algorithm Overview
  12. Proof of Correctness
  13. Probabilistic Analysis
  14. Proof of Proposition \ref{['prop:greedy']}

Key Result

Theorem 1.1

Let $\varepsilon \in (0,1)$ be arbitrary, and let $G$ be an $n$-vertex graph with $m$ edges having maximum degree $\Delta \geq 500\log n / \varepsilon$. There is a randomized sequential algorithm that finds a proper $(1+\varepsilon)\Delta$-edge-coloring of $G$ in time $O\left(m\log^3n/\varepsilon^3\

Figures (6)

  • Figure 1: The process of augmenting a Vizing chain $(F, P)$.
  • Figure 2: The process of augmenting an initial segment of a Vizing chain after flagging an edge.
  • Figure 3: The process of shifting a fan.
  • Figure 4: The process of flipping an $\alpha\beta$-path.
  • Figure 5: The process of flipping $P$ and then shifting $F$ in a Vizing chain $(F, P)$.
  • ...and 1 more figures

Theorems & Definitions (19)

  • Theorem 1.1
  • Definition 1.1: Augmenting subgraphs
  • Definition 2.1: Fans
  • Definition 2.2: Alternating Paths
  • Definition 2.3: Vizing chains
  • Proposition 2.3
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 9 more