Abstract A graph G is perihamiltonian if G itself is non-hamiltonian, yet every edge-contracted subgraph of G is hamiltonian. These graphs form a superclass of the hypohamiltonian graphs. By applying a recent result of Wiener on path-critical graphs, we prove the existence of infinitely many perihamiltonian graphs of connectivity k for any k ≥ 2. We also show that every planar perihamiltonian graph of connectivity k contains a vertex of degree k. This strengthens a theorem of Thomassen, and entails that if in a polyhedral graph of minimum degree at least 4 the set of vertices whose removal yields a non-hamiltonian graph is independent, the graph itself must be hamiltonian. Finally, while we here prove that there are infinitely many polyhedral perihamiltonian graphs containing no adjacent cubic vertices, whether an analogous result holds for the hypohamiltonian case remains open.

中文翻译：

---

非哈密顿图，其中每个边收缩的子图都是哈密顿图

摘要 如果图 G 本身是非哈密顿图，则图 G 是哈密顿图，而 G 的每个边收缩子图都是哈密顿图。这些图形成了亚哈密尔顿图的超类。通过在路径临界图上应用 Wiener 的最新结果，我们证明了对于任意 k ≥ 2 存在无穷多个连通性 k 的近哈密尔顿图。我们还表明，每个连通性 k 的平面近哈密尔顿图都包含一个度为 k 的顶点。这加强了 Thomassen 的定理，并且意味着如果在最小度数至少为 4 的多面体图中，去除产生非哈密顿图的顶点集是独立的，则该图本身必须是哈密顿图。最后，虽然我们在这里证明有无限多个不包含相邻三次顶点的多面体近哈密尔顿图，