An infinite family of minimally nonperfectly divisible graphs with a bisimplicial vertex

Abstract

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.