A linear-time algorithm for $(1+ε)Δ$-edge-coloring

TL;DR

This work delivers a randomized, near-linear time algorithm for obtaining a proper $(1+\varepsilon)\Delta$-edge-coloring for graphs of maximum degree $\Delta\ge 1/\varepsilon$ using $q=(1+\varepsilon)\Delta$ colors. The core method is a refined Multi-Step Vizing Algorithm (MSVA) that assembles non-intersecting Vizing chains from random fans and alternating paths, combined with an entropy-compression analysis to bound total iterations. A key advance is the ability to guarantee small augmenting subgraphs of size $O((\log n)/\varepsilon^4)$ on average and to bound total runtime by $O(m\log(1/\varepsilon)/\varepsilon^4)$ with exponentially small failure probability. The approach removes prior $\,\Delta$-dependence in the running time and extends linear-time colorings to the full range of $\Delta$, though reducing to the $\Delta+1$ palette remains an open challenge. The results have implications for fast constructive edge-coloring and related algorithmic graph coloring questions.

Abstract

We present a randomized algorithm that, given a constant $ε> 0$, outputs a proper $(1+ε)Δ$-edge-coloring of an $m$-edge simple graph $G$ of maximum degree $Δ\geq 1/ε$ in $O(m)$ time with high probability. This is the first linear-time algorithm for this problem covering the full range of possible values of $Δ$. Indeed, even for edge-coloring with $2Δ- 1$ colors (i.e., meeting the "greedy" bound), no such linear-time algorithm has been previously known.

Figures (11)

• Figure 1: A multi-step Vizing chain.
• Figure 2: An iteration of the basic while loop in Algorithm \ref{['inf:MSVC']} when there is an intersection between $F + P$ and $F_j + P_j$.
• Figure 3: Shifting a coloring along a chain (Greek letters represent colors).
• Figure 4: The chain $P(e; \varphi, \alpha\beta)$.
• Figure 5: The process of shifting a fan.

Theorems & Definitions (63)

• Definition 1.1: Proper edge-coloring
• Theorem 1.2: Vizing Vizing
• Theorem 1.3: $(1+\varepsilon)\Delta$-edge-coloring in linear time
• Definition 1.4: Augmenting subgraphs
• Theorem 1.5: Chang--He--Li--Pettie--Uitto CHLPU
• Theorem 1.6: Small augmenting subgraphs for $(1+\varepsilon)\Delta$-edge-coloring
• Definition 2.1: Happy chains
• Definition 2.2: Path chains
• Definition 2.3: Internal vertices and edges
• Definition 2.4: Hopeful and successful edges