A Note on Circular Arc Online Coloring using First Fit

TL;DR

This paper extends online coloring bounds from interval graphs to circular arc graphs, proving that First Fit can color such graphs with at most 9 times their maximum clique size.

Contribution

It provides a new upper bound of 9w(G) for online First Fit coloring of circular arc graphs, adapting techniques beyond interval graphs.

Findings

First Fit colors circular arc graphs with at most 9w(G) colors.

The previous bound for interval graphs was 8w(G).

A new proof technique is introduced for circular arc graphs.

Abstract

In Raman (2007), using a column construction technique it is proved that every interval graph can be colored online with First Fit with at most $8w(G)$ colors, where $w(G)$ is the size of the maximum clique of $G$. Since the column construction can not be adapted to circular arc graphs we give a different proof to establish an upper bound of $9w(G)$ for online coloring a circular arc graph $G$ with the First Fit algorithm.