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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. 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N. Ellingham, Pouria Salehi Nowbandegani, Songling Shan","submitted_at":"2019-01-07T18:25:39Z","abstract_excerpt":"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. 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