Online Edge Coloring via Tree Recurrences and Correlation Decay

TL;DR

This paper presents an online algorithm for edge coloring that achieves near-optimal coloring in high-degree graphs, using tree recurrence techniques to handle correlation decay in a novel way.

Contribution

It introduces a new online edge coloring algorithm for high-degree graphs that leverages tree recurrences and correlation decay, advancing the conjecture of Bar-Noy, Motwani, and Naor.

Findings

Achieves $(e/(e-1)+o(1)) imes ext{max degree}$ edge coloring online.

Works against oblivious adversaries in graphs with degree $ ext{omega}( ext{log } n)$.

Reduces the problem to a matching problem on locally treelike graphs.

Abstract

We give an online algorithm that with high probability computes a $\left(\frac{e}{e-1} + o(1)\right)\Delta$ edge coloring on a graph $G$ with maximum degree $\Delta = \omega(\log n)$ under online edge arrivals against oblivious adversaries, making first progress on the conjecture of Bar-Noy, Motwani, and Naor in this general setting. Our algorithm is based on reducing to a matching problem on locally treelike graphs, and then applying a tree recurrences based approach for arguing correlation decay.