完美可分性、线性可分性与无椅图
Perfect Divisibility, Linear Divisibility and Chair-Free Graphs
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中文总结 AI 辅助
本文研究完美可分性与线性可分性，构造无限多反例证明完美可分图的团数相关色数界的逆不成立，证明无椅图为(2,2)-线性可分图，且独立得到该结果。
中文摘要 AI 辅助
若一个图的每个至少含一条边的诱导子图都能划分为一个完美诱导子图和一个团数更小的诱导子图，则称该图是完美可分的。每个完美可分图G对其每个诱导子图H都满足χ(H)≤C(ω(H)+1,2)。我们证明其逆命题不成立：对每个非负整数t，图P(17)∨K_t对其每个诱导子图都满足该界，但它不是完美可分的，由此得到无限多个反例。受此区分启发，我们引入(k,ℓ)-线性可分性，并证明每个(k,ℓ)-线性可分图G满足χ(G)≤k·C(ω(G)+1,2)。作为该框架的应用，我们给出直接的结构分解，证明每个无椅图都是(2,2)-线性可分的，其中椅图是将K_{1,3}的一条边各细分一次得到的图。我们在得知Liu、Sun、Wang、Wu和Zeng的最新预印本[arXiv:2608.13519]之前独立得到了该无椅图结果，该预印本证明了更强的结论：每个无椅图都是完美权重可分的，因此满足χ(G)≤C(ω(G)+1,2)。
英文摘要
A graph is perfectly divisible if every induced subgraph with at least one edge admits a partition into a perfect induced subgraph and an induced subgraph with smaller clique number. Every perfectly divisible graph $G$ satisfies $χ(H)\leq\binom{ω(H)+1}{2}$ for every induced subgraph $H$ of $G$. We show that the converse fails: for every non-negative integer $t$, the graph $P(17)\vee K_t$ satisfies this bound for every induced subgraph but is not perfectly divisible, yielding an infinite family of counterexamples. Motivated by this distinction, we introduce $(k,\ell)$-linear divisibility and prove that every $(k,\ell)$-linearly divisible graph $G$ satisfies $χ(G)\leq k\binom{ω(G)+1}{2}$. As an application of this framework, we give a direct structural decomposition showing that every chair-free graph is $(2,2)$-linearly divisible, where a chair is obtained from $K_{1,3}$ by subdividing one edge once. This chair-free result was obtained independently before we became aware of a recent preprint of Liu, Sun, Wang, Wu, and Zeng [arXiv:2608.13519], who prove the stronger statement that every chair-free graph is perfectly weight divisible and hence satisfies $χ(G)\leq\binom{ω(G)+1}{2}$.
