The relation between Hamiltonian and $1$-tough properties of the Cartesian product graphs

The relation between Hamiltonicity and toughness of a graph is a long standing research problem. The paper studies the Hamiltonicity of the Cartesian product graph $G_1square G_2$ of graphs $G_1$ and $G_2$ satisfying that $G_1$ is traceable and $G_2$ is connected with a path factor. Let Pn be the path of order $n$ and $H$ be a connected bipartite graph. With certain requirements of $n$, we show that the following three statements are equivalent: (i) $P_nsquare H$ is Hamiltonian; (ii) $P_nsquare H$ is $1$-tough; and (iii) $H$ has a path factor.