Linear colorings of graphs

TL;DR

This paper explores the properties of the linear chromatic number in graphs, providing improved bounds for various graph classes and examining its relationship with treedepth, motivated by algorithmic applications.

Contribution

It advances understanding of linear chromatic number by establishing new bounds and analyzing its relation to treedepth across different graph classes.

Findings

Linear chromatic number is polynomially related to treedepth.

Improved bounds for linear chromatic number in specific graph classes.

Support for the conjecture relating treedepth and linear chromatic number.

Abstract

Motivated by algorithmic applications, Kun, O'Brien, Pilipczuk, and Sullivan introduced the parameter linear chromatic number as a relaxation of treedepth and proved that the two parameters are polynomially related. They conjectured that treedepth could be bounded from above by twice the linear chromatic number. In this paper we investigate the properties of linear chromatic number and provide improved bounds in several graph classes.