The \emph{prism} over a graph $G$ is the product $G \Box K_2$, i.e., the graph obtained by taking two copies of $G$ and adding a perfect matching joining the two copies of each vertex by an edge. The graph $G$ is called \emph{prism-hamiltonian} if it has a hamiltonian prism. Jung showed that every $1$-tough $P_4$-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for $k \geq 1$ a $P_4$-free graph has a spanning \emph{$k$-walk} (closed walk using each vertex at most $k$ times) if and only if it is $\frac{1}{k}$-tough. As our main result, we show that for the class of $P_4$-free graphs, the three properties of being prism-hamiltonian, having a spanning $2$-walk, and being $\frac{1}{2}$-tough are all equivalent.

中文翻译：

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无 P4 图的韧性和棱柱哈密性

图 $G$ 上的 \emph{prism} 是乘积 $G \Box K_2$，即通过取 $G$ 的两个副本并添加一个完美匹配获得的图，通过边将每个顶点的两个副本连接起来. 如果图 $G$ 具有哈密顿棱镜，则称为 \emph{prism-hamiltonian}。荣格证明了每一个具有至少三个顶点的 $1$-tough $P_4$-free 图都是汉密尔顿图。在本文中，我们将其扩展为观察到对于 $k \geq 1$，无 $P_4$ 的图具有跨越 \emph{$k$-walk}（使用每个顶点至多 $k$ 次的封闭步行），如果并且仅当它是 $\frac{1}{k}$-tough 时。作为我们的主要结果，我们证明对于无 $P_4$ 的图类，具有棱柱哈密顿分布、跨越 $2$-walk 和 $\frac{1}{2}$-tough 的三个性质都是等价的。