Špacapan recently showed that there exist 3-polytopes with non-Hamiltonian prisms, disproving a conjecture of Rosenfeld and Barnette. By adapting Špacapan's approach we strengthen his result in several directions. We prove that there exists an infinite family of counterexamples to the Rosenfeld–Barnette conjecture, each member of which has maximum degree 37, is of girth 4, and contains no odd-length face with length less than $k$ for a given odd integer $k$. We also show that for any given 3-polytope $H$ there is a counterexample containing $H$ as an induced subgraph. This yields an infinite family of non-Hamiltonian 4-polytopes in which the proportion of quartic vertices tends to 1. However, Barnette's conjecture stating that every 4-polytope in which all vertices are quartic is Hamiltonian still stands. Finally, we prove that the Grünbaum–Walther shortness coefficient of the family of all prisms of 3-polytopes is at most 59/60.

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关于具有非哈密尔顿棱镜的 3-polytopes

Špacapan 最近表明存在具有非汉密尔顿棱镜的 3-polytopes，反驳了 Rosenfeld 和 Barnette 的猜想。通过调整 Špacapan 的方法，我们在几个方向上加强了他的结果。我们证明了 Rosenfeld-Barnette 猜想存在无限的反例族，其中每个成员的最大度数为 37，周长为 4，并且不包含长度小于$克$ 对于给定的奇数 $克$. 我们还表明，对于任何给定的 3-polytope$H$ 有一个反例包含 $H$作为诱导子图。这产生了一个无限的非哈密尔顿 4-polytope 族，其中四次顶点的比例趋向于 1。然而，Barnette 的猜想表明，其中所有顶点都是四次的每个 4-polytope 都是哈密顿量的猜想仍然成立。最后，我们证明了所有 3-polytopes 棱柱族的 Grünbaum-Walther 短度系数至多为 59/60。