SIAM Journal on Discrete Mathematics, Volume 35, Issue 3, Page 1706-1728, January 2021.
Motivated by a conjecture of Grünbaum and a problem of Katona, Kostochka, Pach, and Stechkin, both dealing with non-Hamiltonian $n$-vertex graphs and their $(n-2)$-cycles, we investigate $K_2$-Hamiltonian graphs, i.e., graphs in which the removal of any pair of adjacent vertices yields a Hamiltonian graph. In this first part, we prove structural properties and show that there exist infinitely many cubic non-Hamiltonian $K_2$-Hamiltonian graphs, both of the 3-edge-colorable and the non-3-edge-colorable variety. In fact, cubic $K_2$-Hamiltonian graphs with chromatic index 4 (such as Petersen's graph) are a subset of the critical snarks. On the other hand, it is proven that non-Hamiltonian $K_2$-Hamiltonian graphs of any maximum degree exist. Several operations conserving $K_2$-Hamiltonicity are described, one of which leads to the result that there exists an infinite family of non-Hamiltonian $K_2$-Hamiltonian graphs in which, asymptotically, a quarter of vertices has the property that removing such a vertex yields a non-Hamiltonian graph. We extend a celebrated result of Tutte by showing that every planar $K_2$-Hamiltonian graph with minimum degree at least 4 is Hamiltonian. Finally, we investigate $K_2$-traceable graphs and discuss open problems.

中文翻译：

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$K_2$-哈密顿图：I

SIAM 离散数学杂志，第 35 卷，第 3 期，第 1706-1728 页，2021 年 1 月。
受 Grünbaum 猜想和 Katona、Kostochka、Pach 和 Stechkin 问题的启发，它们都处理非汉密尔顿 $n$-顶点图及其 $(n-2)$-循环，我们研究 $K_2$-Hamiltonian图，即去除任意一对相邻顶点产生哈密顿图的图。在第一部分中，我们证明了结构性质并证明存在无限多个三次非哈密顿 $K_2$-哈密尔顿图，包括 3 边可着色和非 3 边可着色变体。事实上，色度指数为 4 的三次 $K_2$-Hamiltonian 图（例如 Petersen 的图）是临界 snarks 的子集。另一方面，证明了任何最大次数的非Hamiltonian$K_2$-Hamiltonian图都存在。描述了几个保存 $K_2$-Hamiltonicity 的操作，其中一个导致的结果是存在一个无限的非汉密尔顿 $K_2$-汉密尔顿图族，其中，渐近地，四分之一的顶点具有去除这样一个顶点会产生非汉密尔顿图的性质。我们通过证明每个最小度数至少为 4 的平面 $K_2$-Hamiltonian 图是哈密顿图来扩展 Tutte 的著名结果。最后，我们研究 $K_2$-traceable 图并讨论开放问题。