无B图猜想的一个证明
A Proof of the B-Free Graphs Conjecture
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中文总结 AI 辅助
该论文证明了无B图猜想，通过Hall型交换论证、核心-冠匹配引理等方法，将问题简化为21个二元可行性系统并验证其不可行性，确立了相关图论不等式。
中文摘要 AI 辅助
令$\boldsymbol{\textit{B}}$为包含6顶点二分图（具有完美匹配）及其补图的类。本文证明，每个无$\boldsymbol{\textit{B}}$图$G$都满足$\boldsymbol{\textit{\u03b1}}(G)+\boldsymbol{\textit{\u03c9}}(G)\u2265 |V(G)|-1$，从而确立了Litjens、Polak和Sivaraman提出的猜想3.1（无B图猜想）。对于最小反例，Hall型交换论证表明，两个最大稳定集以及两个最大团在至多两个顶点上不同；随后，核心-冠匹配引理强制$\boldsymbol{\textit{|}}\boldsymbol{\textit{\u03b1}}(G)-\boldsymbol{\textit{\u03c9}}(G)\boldsymbol{\textit{|}}\u2264 2$。通过对适当稠密和稀疏顶点的双重计数，问题被简化为21个至多14顶点的二元可行性系统，其不可行性通过两种独立的精确编码验证，同时对禁族约束进行单独的穷尽式验证。
英文摘要
Let $\mathcal{B}$ be the class consisting of the six-vertex bipartite graphs that possess a perfect matching and their complements. It is proved that every $\mathcal{B}$-free graph $G$ satisfies $α(G)+ω(G)\ge |V(G)|-1$. This establishes Conjecture 3.1 of Litjens, Polak and Sivaraman (B-Free Graphs Conjecture). For a smallest counterexample, Hall-type exchange arguments show that two maximum stable sets, and likewise two maximum cliques, differ in at most two vertices. A core-corona matching lemma then forces $|α(G)-ω(G)|\le 2$. Double counting between suitably dense and sparse vertices reduces the problem to twenty-one binary feasibility systems on at most fourteen vertices. Their infeasibility is verified by two independent exact encodings, with a separate exhaustive validation of the forbidden-family constraints.
