Linear Versus Centred Colouring via Pseudogrids

Abstract

A centred colouring of a graph is a vertex colouring in which every connected subgraph contains a vertex whose colour is unique and a linear colouring is a vertex colouring in which every (not-necessarily induced) path contains a vertex whose colour is unique. For a graph $G$, the centred chromatic number ${\chi }_{\text{cen}}\left(G\right)$ and the linear chromatic number ${\chi }_{\text{lin}}\left(G\right)$ denote the minimum number of distinct colours required for a centred, respectively, linear colouring of $G$. From these definitions, it follows immediately that ${\chi }_{\text{lin}}\left(G\right)\le {\chi }_{\text{cen}}\left(G\right)$ for every graph $G$. The centred chromatic number is equivalent to treedepth and has been studied extensively. Much less is known about linear colouring. Kun and colleagues prove that ${\chi }_{\text{cen}}\left(G\right)\le \tilde{O}\left({\chi }_{\text{lin}}{\left(G\right)}^{190}\right)$ for any graph $G$ and conjecture that ${\chi }_{\text{cen}}\left(G\right)\le 2{\chi }_{\text{lin}}\left(G\right)$. Their upper bound was subsequently improved by Czerwiński and colleagues to ${\chi }_{\text{cen}}\left(G\right)\le \tilde{O}\left({\chi }_{\text{lin}}{\left(G\right)}^{19}\right)$. The proof of both upper bounds relies on establishing a lower bound on the linear chromatic number of pseudogrids, which appear in the proof due to their critical relationship to treewidth. Specifically, Kun and colleagues prove that $k\times k$ pseudogrids have linear chromatic number $\Omega \left(\sqrt{k}\right)$. Our main contribution is establishing a tight bound on the linear chromatic number of pseudogrids, specifically ${\chi }_{\text{lin}}\left(G\right)\ge \Omega \left(k\right)$ for every $k\times k$ pseudogrid $G$. As a consequence we improve the general bound for all graphs to ${\chi }_{\text{cen}}\left(G\right)\le \tilde{O}\left({\chi }_{\text{lin}}{\left(G\right)}^{10}\right)$. In addition, this tight bound gives further evidence in support of Kun and colleagues' conjecture (above) that the centred chromatic number (i.e., the treedepth) of any graph is upper bounded by a linear function of its linear chromatic number.