Linear versus centred chromatic numbers

TL;DR

This work tightens the relationship between centred and linear chromatic numbers by proving a near-linear lower bound for the linear chromatic number of $k\times k$ pseudogrids, which in turn yields a $\tilde{O}(\chi_{\mathrm{lin}}(G)^{10})$ upper bound on the centred chromatic number for all graphs. The authors develop a detailed grid-pseudogrid framework, introduce interior colour-frequency controls, and leverage a sequence of combinatorial and probabilistic tools—including Hall's theorem and the Lovász Local Lemma—to construct well-separated colour representatives and a path that violates linear colouring unless $\chi_{\mathrm{lin}}(G)$ is large. The result strengthens the evidence for a linear bound $\chi_{\mathrm{cen}}(G) \le 2\chi_{\mathrm{lin}}(G)$ and advances the understanding of how grid minors and treewidth influence chromatic parameters. The techniques, notably the packing lemmas and path-construction through well-separated pairs, offer a robust toolkit for relating treedepth/width-type parameters to colouring properties in complex graph families.

Abstract

$\DeclareMathOperator{\chicen}{χ_{\mathrm{cen}}}\DeclareMathOperator{\chilin}{χ_{\mathrm{lin}}}$ 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 $Ω(\sqrt{k})$. Our main contribution is establishing a tight bound on the linear chromatic number of pseudogrids, specifically $\chilin(G)\ge Ω(k)$ 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.

Figures (7)

• Figure 1: A $6\times 4$ pseudogrid. Black vertices are contained in $P_v$ for some vertex $v$ of $G_{6\times 4}$ and white vertices are contained in $P_{vw}$ for some edge of $G_{6\times 4}$. Colour are used to distinguish between cases (Q1) (blue), (Q2) (pink), and (Q3) (gold).
• Figure 2: Constructing the sets $S_1$ and $S_2$ and the path $P$.
• Figure 3: Some examples of $3$- and $4$-element vertex subsets that cannot all be contained in a single path (that must also contain other vertices not pictured).
• Figure 4: The proof of \ref{['make_disjoint']}. Disks are vertices in $S$ and crosses are the resulting elements of $X$. The large yellow boxes are $B_{r-1}(v)$ for some $v\in S$ and the small mauve and green boxes are $B_{p+1}(x)$ and $B_{p}(x)$, respectively, for some $x\in X$.
• Figure 5: A simpler solution for colourings of the grid. The (orange) tree used to create the (purple) Hamiltonian path is created so that its edges and leaves use only odd-numbered rows and columns of the dual graph.

Theorems & Definitions (22)

• Theorem 1: kun.obrien.ea:polynomialczerwinski.nadara.ea:improved
• Conjecture 2: kun.obrien.ea:polynomial
• Lemma 3
• Theorem 4
• Theorem 5: hall:on
• Corollary 6
• Lemma 7
• Lemma 8
• proof
• Lemma 9