Online Coloring for Graphs of Large Odd Girth

TL;DR

This paper presents a new deterministic online coloring algorithm for graphs with large odd girth, achieving sublinear color bounds for any small positive exponent.

Contribution

It introduces a method to color graphs with large odd girth online using $O(n^{ ext{small } ext{exponent}})$ colors, improving previous bounds for certain girth values.

Findings

Deterministic online coloring with $O(n^{ ext{small } ext{exponent}})$ colors for graphs with large odd girth.

Existence of a girth threshold $g'$ where the new coloring bound applies.

Improvement over the previous $O(n^{1/2})$ coloring algorithm for graphs with odd girth at least 7.

Abstract

We study the problem of online coloring for graphs with large odd girth. The best previously known algorithm uses $O(n^{1/2})$ colors, which was discovered by Kierstead in 1998. This algorithm works when the odd girth is 7 or more. In this paper, we provide the following: for every $\varepsilon > 0$, there exists a constant $g' \in \{3, 5, 7, \dots\}$ such that graphs with odd girth at least $g'$ can be deterministically colored online using $O(n^{\varepsilon})$ colors.