AI 中文总结
反驳Hoàng关于极小非完全可分图不能含双单纯顶点的猜想，构造反例图，基于带根图用强制机制，其无完美划分但真诱导子图可分，还对Hu、Xu和Zhuang的规定顶点问题给出否定回答。
AI 中文摘要
我们反驳了Hoàng的猜想，即极小非完全可分图不能包含双单纯顶点。我们的反例有93个顶点、320条边、团数为3，以及一个度数为4的双单纯顶点。图\(H\)的完美划分是将\(V(H)\)划分为\(A\)和\(B\)，使得\(H[A]\)是完美的且\(\omega(H[B]) < \omega(H)\)；当图的每个至少有一条边的诱导子图都有这样的划分时，该图是完全可分的。所构造的图没有完美划分，但其每个真诱导子图都是完全可分的。其构造基于带根图使用了一种强制机制：六个15顶点的带根图将它们的根强制放入\(A\)，然后一个九顶点的辅助图将一个三角形强制放入\(B\)。我们在一个带根合成引理中分离出这种机制，该引理通过符号证明每个真诱导子图都是完全可分的。所有关于带根图和辅助图的有限断言都通过精确的穷举计算进行了验证，并且有一个独立的实现提供了交叉检查。同样的构造对Hu、Xu和Zhuang的一个规定顶点问题给出了否定答案。
英文摘要
We disprove Hoàng's conjecture that a minimally nonperfectly divisible graph cannot contain a bisimplicial vertex by constructing an explicit infinite family. For every integer $t\geq1$, the graph $G_t$ in this family has clique number three, contains a bisimplicial vertex of degree four, and satisfies \[ |V(G_t)|=93+30(t-1),\qquad |E(G_t)|=320+104(t-1). \] In particular, the members are pairwise nonisomorphic. The construction uses a fixed 15-vertex rooted graph and a variable auxiliary graph. Every copy of the rooted graph forces its identified root into the perfect part of every perfect division. Three induced odd holes in the auxiliary graph then force a triangle into the other part. A uniform assignment lemma and a rooted product lemma show symbolically that every proper induced subgraph of every $G_t$ is perfectly divisible. The finite properties of the fixed rooted graph are verified by exact exhaustive computation, with an independent implementation providing a cross-check. The construction also gives an infinite family of negative examples to a prescribed-vertex problem of Hu, Xu and Zhuang.
Comments11 pages, 1 figure; ancillary files include exact verification code and reproducibility data
