AI 中文总结
该研究证明图的中心着色最少颜色数与线性着色最少颜色数之间存在超线性分离，构造由OpenAI的GPT-5.6 Sol Pro完成。
AI 中文摘要
图的顶点着色若满足每个连通子图都有一个颜色唯一的顶点，则称为中心着色；若每个路径都有一个颜色唯一的顶点，则称为线性着色。设χ_cen(G)和χ_lin(G)分别为图G的中心着色和线性着色所需的最少颜色数。本文给出一组图，证明若存在非递减函数f使得对所有图G有χ_cen(G)≤f(χ_lin(G))，则f(k)=Ω(k²/log k)，该构造由OpenAI的GPT-5.6 Sol Pro完成。
英文摘要
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 $χ_{\mathrm{cen}}(G)$ and $χ_{\mathrm{lin}}(G)$ 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 $χ_{\mathrm{cen}}(G) \leq f(χ_{\mathrm{lin}}(G))$ for every graph $G$, then $f(k) = Ω(k^2 / \log k)$. The construction was found by OpenAI's GPT-5.6 Sol Pro.
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