# Tight Bounds for Online Edge Coloring

## TL;DR

This paper advances online edge coloring by providing near-optimal algorithms for bipartite graphs with high maximum degree, resolving a long-standing conjecture for adversarial arrivals and introducing novel fractional relaxation techniques.

## Contribution

It introduces a $(1+o(1))\Delta$-edge-coloring algorithm for adversarial arrivals in bipartite graphs with high degree, and establishes optimal bounds for unknown $\Delta$ scenarios.

## Key findings

- Achieved a $(1+o(1))\Delta$-edge-coloring algorithm for $\Delta= ext{omega}( ext{log} ext{n})$
- Proved no $(rac{e}{e-1}- ext{Omega}(1))\Delta$-edge-coloring algorithm exists without prior knowledge of $\Delta$
- Developed a fractional relaxation and near-lossless rounding scheme for online edge coloring.

## Abstract

Vizing's celebrated theorem asserts that any graph of maximum degree $\Delta$ admits an edge coloring using at most $\Delta+1$ colors. In contrast, Bar-Noy, Naor and Motwani showed over a quarter century that the trivial greedy algorithm, which uses $2\Delta-1$ colors, is optimal among online algorithms. Their lower bound has a caveat, however: it only applies to low-degree graphs, with $\Delta=O(\log n)$, and they conjectured the existence of online algorithms using $\Delta(1+o(1))$ colors for $\Delta=\omega(\log n)$. Progress towards resolving this conjecture was only made under stochastic arrivals (Aggarwal et al., FOCS'03 and Bahmani et al., SODA'10). We resolve the above conjecture for \emph{adversarial} vertex arrivals in bipartite graphs, for which we present a $(1+o(1))\Delta$-edge-coloring algorithm for $\Delta=\omega(\log n)$ known a priori. Surprisingly, if $\Delta$ is not known ahead of time, we show that no $\big(\frac{e}{e-1} - \Omega(1) \big) \Delta$-edge-coloring algorithm exists. We then provide an optimal, $\big(\frac{e}{e-1}+o(1)\big)\Delta$-edge-coloring algorithm for unknown $\Delta=\omega(\log n)$. Key to our results, and of possible independent interest, is a novel fractional relaxation for edge coloring, for which we present optimal fractional online algorithms and a near-lossless online rounding scheme, yielding our optimal randomized algorithms.