Linear colorings of graphs
Claire Hilaire, Matjaž Krnc, Martin Milanič, Jean‐Florent Raymond · European Journal of Combinatorics · 2026
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.