AI 中文总结
研究满足特定条件的图的可拆分性，核心方法是证明不含\(C_4\)且每个诱导子图有双单纯顶点的图是\((s,t)\)-可拆分的，结合相关结果，主要贡献是证明埃尔德什 - 洛瓦兹蒂哈尼猜想对所有不含偶洞的图成立。
AI 中文摘要
设\(s,t\geq2\)为整数。若图\(G\)的顶点集\(V(G)\)能被划分为两个集合\(S\)和\(T\)，使得\(\chi(G[S])\geq s\)且\(\chi(G[T])\geq t\)，则称\(G\)是\((s,t)\)-可拆分的。1968年的埃尔德什 - 洛瓦兹蒂哈尼猜想断言，每个满足\(\omega(G)<\chi(G)=s + t - 1\)的图\(G\)是\((s,t)\)-可拆分的。若图中一个顶点的邻域集可表示为两个团的并集，则该顶点是双单纯的。我们证明，若图\(G\)不包含\(C_4\)作为诱导子图且其每个诱导子图都有一个双单纯顶点，那么\(G\)是\((s,t)\)-可拆分的。结合我们的结果与Chudnovsky和Seymour最近的结果（即每个非空不含偶洞的图都有一个双单纯顶点），我们得出埃尔德什 - 洛瓦兹蒂哈尼猜想对所有不含偶洞的图都成立。
英文摘要
Let $s, t\ge2 $ be integers. A graph $G$ is $(s,t)$-splittable if $V(G)$ can be partitioned into two sets $S$ and $T$ such that $χ(G[S ]) \ge s$ and $χ(G[T ]) \ge t$. The Erdős-Lovász Tihany Conjecture from 1968 asserts that every graph $G$ satisfying $ω(G)<χ(G)=s+t-1$ is $(s,t)$-splittable. A vertex of a graph is bisimplicial if the set of its neighbors can be expressed as the union of two cliques. Let $G$ be a graph with $ω(G)<χ(G)=s+t-1$. We prove that if $G$ does not contain $C_4$ as an induced subgraph and every induced subgraph of $G$ has a bisimplicial vertex, then $G$ is $(s,t)$-splittable. Combining our result with a recent result of Chudnovsky and Seymour, which states that every non-empty even-hole-free graph has a bisimplicial vertex, we obtain that the Erdős-Lovász Tihany Conjecture holds for all even-hole-free graphs.
