$\DeclareMathOperator{\chicen}{\chi_{\mathrm{cen}}}\DeclareMathOperator{\chilin}{\chi_{\mathrm{lin}}}$ A centred colouring of a graph is a vertex colouring in which every connected subgraph contains a vertex whose colour is unique and a \emph{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 $\chicen(G)$ and the linear chromatic number $\chilin(G)$ denote the minimum number of distinct colours required for a centred, respectively, linear colouring of $G$. From these definitions, it follows immediately that $\chilin(G)\le \chicen(G)$ 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 et al [Algorithmica 83(1)] prove that $\chicen(G) \le \tilde{O}(\chilin(G)^{190})$ for any graph $G$ and conjecture that $\chicen(G)\le 2\chilin(G)$. Their upper bound was subsequently improved by Czerwinski et al [SIDMA 35(2)] to $\chicen(G)\le\tilde{O}(\chilin(G)^{19})$. 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 et al prove that $k\times k$ pseudogrids have linear chromatic number $\Omega(\sqrt{k})$. Our main contribution is establishing a tight bound on the linear chromatic number of pseudogrids, specifically $\chilin(G)\ge \Omega(k)$ for every $k\times k$ pseudogrid $G$. As a consequence we improve the general bound for all graphs to $\chicen(G)\le \tilde{O}(\chilin(G)^{10})$. In addition, this tight bound gives further evidence in support of Kun et al's conjecture (above) that the centred chromatic number of any graph is upper bounded by a linear function of its linear chromatic number.

翻译：设 $\DeclareMathOperator{\chicen}{\chi_{\mathrm{cen}}}\DeclareMathOperator{\chilin}{\chi_{\mathrm{lin}}}$ 图的中心染色是一种顶点染色方式，要求每个连通子图中存在一个颜色唯一的顶点；而**线性染色**则要求每条（未必导出的）路中均存在颜色唯一的顶点。对于图 $G$，中心色数 $\chicen(G)$ 和线性色数 $\chilin(G)$ 分别表示对 $G$ 进行中心染色和线性染色所需的最小颜色数。由定义直接可得，对任意图 $G$ 有 $\chilin(G)\le \chicen(G)$。中心色数等价于树深度（treedepth），已被广泛研究；而线性染色的研究则相对较少。Kun 等人 [Algorithmica 83(1)] 证明了对任意图 $G$ 有 $\chicen(G) \le \tilde{O}(\chilin(G)^{190})$，并猜想 $\chicen(G)\le 2\chilin(G)$。其后 Czerwinski 等人 [SIDMA 35(2)] 将该上界改进为 $\chicen(G)\le\tilde{O}(\chilin(G)^{19})$。上述两个上界的证明均依赖于对伪网格图（pseudogrids）线性色数下界的建立——由于伪网格与树宽（treewidth）的关键关联，该结构在证明中起到核心作用。具体而言，Kun 等人证明了 $k\times k$ 伪网格图的线性色数为 $\Omega(\sqrt{k})$。本文的主要贡献在于建立伪网格图线性色数的紧界：对任意 $k\times k$ 伪网格图 $G$，有 $\chilin(G)\ge \Omega(k)$。基于此结果，我们将所有图的通用上界改进为 $\chicen(G)\le \tilde{O}(\chilin(G)^{10})$。此外，该紧界进一步支持了 Kun 等人的上述猜想，即任意图的中心色数均可被其线性色数的线性函数上界所控制。