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Edge-coloring sparse graphs with $Δ$ colors in quasilinear time

Lukasz Kowalik

TL;DR

This work achieves Δ-edge-coloring for graphs with bounded maximum average degree under the condition Δ ≥ 2 mad(G), delivering quasilinear-time randomized (O(m mad(G)^3 log n)) and deterministic (O(m mad(G)^7 log n)) algorithms. The approach blends Vizing Adjacency Lemma with a careful analysis of weak edges, utilizing rotating fans and alternating paths, and introduces a partitioning technique to separate subproblems, enabling disjoint color palettes and merging. A key theoretical contribution is a linear lower bound on the number of Δ-weak edges via discharging, and the algorithm derandomizes prior quasi-linear results for bounded mad graphs. The results extend to corollaries such as a deterministic quasi-linear Δ+1 coloring and open up directions for relaxing sparsity constants, dynamic updates, and LOCAL-model implementations.

Abstract

In this paper we show that every graph $G$ of bounded maximum average degree ${\rm mad}(G)$ and with maximum degree $Δ$ can be edge-colored using the optimal number of $Δ$ colors in quasilinear time, whenever $Δ\ge 2{\rm mad}(G)$. The maximum average degree is within a multiplicative constant of other popular graph sparsity parameters like arboricity, degeneracy or maximum density. Our algorithm extends previous results of Chrobak and Nishizeki [J. Algorithms, 1990] and Bhattacharya, Costa, Panski and Solomon [ESA 2024].

Edge-coloring sparse graphs with $Δ$ colors in quasilinear time

TL;DR

This work achieves Δ-edge-coloring for graphs with bounded maximum average degree under the condition Δ ≥ 2 mad(G), delivering quasilinear-time randomized (O(m mad(G)^3 log n)) and deterministic (O(m mad(G)^7 log n)) algorithms. The approach blends Vizing Adjacency Lemma with a careful analysis of weak edges, utilizing rotating fans and alternating paths, and introduces a partitioning technique to separate subproblems, enabling disjoint color palettes and merging. A key theoretical contribution is a linear lower bound on the number of Δ-weak edges via discharging, and the algorithm derandomizes prior quasi-linear results for bounded mad graphs. The results extend to corollaries such as a deterministic quasi-linear Δ+1 coloring and open up directions for relaxing sparsity constants, dynamic updates, and LOCAL-model implementations.

Abstract

In this paper we show that every graph of bounded maximum average degree and with maximum degree can be edge-colored using the optimal number of colors in quasilinear time, whenever . The maximum average degree is within a multiplicative constant of other popular graph sparsity parameters like arboricity, degeneracy or maximum density. Our algorithm extends previous results of Chrobak and Nishizeki [J. Algorithms, 1990] and Bhattacharya, Costa, Panski and Solomon [ESA 2024].
Paper Structure (11 sections, 15 theorems, 20 equations, 1 figure, 1 table, 1 algorithm)

This paper contains 11 sections, 15 theorems, 20 equations, 1 figure, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Algorithms for general graphs
  3. Algorithms for graph classes
  4. Uniformly sparse graphs and our result
  5. Sketch of our approach
  6. Organization of the paper
  7. Lower bound for the number of weak edges
  8. Algorithmic Vizing Adjacency Lemma
  9. The main algorithm
  10. Conclusion and further research
  11. High probability guarantee for the running time

Key Result

Theorem 1

Every graph $G$ with $n$ vertices and $m$ edges such that $\Delta(G)\ge 2\mathop{\mathrm{mad}}\nolimits(G)$ can be $\Delta(G)$-edge-colored

Figures (1)

  • Figure 1: Rotating a fan. The numbers next to vertices denote free colors.

Theorems & Definitions (30)

  • Theorem 1
  • Theorem 2: Vizing Adjacency Lemma, VAL vizing:critical
  • Theorem 3
  • proof
  • Claim 1
  • Claim 2
  • Claim 3
  • Theorem 4
  • proof
  • Lemma 5
  • ...and 20 more