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arXiv · 2608.18620

Superlinear separation between linear and centered colorings

Jędrzej Hodor·Piotr Micek
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Abstract

A vertex-coloring of a graph is centered if every connected subgraph has a vertex with a unique color. A vertex-coloring of a graph is linear if every path in the graph has a vertex with a unique color. Let $\chi_{\mathrm{cen}}(G)$ and $\chi_{\mathrm{lin}}(G)$ be the minimum number of colors in a centered (resp. linear) coloring of $G$. We present a family of graphs witnessing that if $f$ is a nondecreasing function such that $\chi_{\mathrm{cen}}(G) \leq f(\chi_{\mathrm{lin}}(G))$ for every graph $G$, then $f(k) = \Omega(k^2 / \log k)$. The construction was found by OpenAI's GPT-5.6 Sol Pro.

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Jędrzej Hodor, Piotr Micek. 2026-08-19. Superlinear separation between linear and centered colorings. https://arxiv.org/abs/2608.18620

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