Abstract
The prism over a graph G is the product G□K2, 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 prism-hamiltonian if it has a hamiltonian prism. Jung showed that every 1-tough P4-free graph with at least three vertices is hamiltonian. In this paper, we extend this to observe that for k≥1 a P4-free graph has a spanning k-walk (closed walk using each vertex at most k times) if and only if it is 1k-tough. As our main result, we show that for the class of P4-free graphs, the three properties of being prism-hamiltonian, having a spanning 2-walk, and being 12-tough are all equivalent.
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