无长诱导路径的图的着色
Coloring graphs with no long induced path
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中文总结 AI 辅助
本文改进了无长诱导路径图的着色上界，提出更紧的色数界并给出基于Gyárfás路径论证的证明。
中文摘要 AI 辅助
设$P_t$表示有$t$个顶点的诱导路径。设$\omega(G)$表示图$G$中一个团的最大顶点数。此前Gyárfás（1987）证明了每个$P_t$-free图$G$满足$\chi(G)\le(t-1)^{\omega(G)-1}$，而Gravier、Hoàng和Maffray（2003）将其改进为$t\ge4$时$\chi(G)\le (t-2)^{\omega(G)-1}$。我们证明对于$t\ge5$，每个$P_t$-free图$G$满足$\chi(G)<c_t\lambda_t^{\omega(G)-1}$，其中$\lambda_t=\tfrac12\bigl(t-2+\sqrt{t(t-4)}\bigr)<t-2$且当$t\to\infty$时$c_t=\sqrt{1+4/(t(t-4))}=1+O(t^{-2})$。该证明基于Gyárfás路径论证的改进，并由Anthropic的Claude Fable 5.1发现。
英文摘要
Let $P_t$ denote the induced path on $t$ vertices. Let $ω(G)$ denote the maximum number of vertices in a clique of a graph $G$. Gyárfás (1987) proved that every $P_t$-free graph $G$ satisfies $χ(G)\le(t-1)^{ω(G)-1}$, and Gravier, Hoàng, and Maffray (2003) improved this to $χ(G)\le (t-2)^{ω(G)-1}$ for $t\ge4$. We lower the base of the exponential by one: for every $t\ge5$, every $P_t$-free graph $G$ satisfies \[ χ(G)\le 3\,(t-3)^{ω(G)+4}. \] The proof combines two refinements of the Gyárfás path argument and was developed with the assistance of Claude Fable 5.1 of Anthropic and GPT Pro of OpenAI.
