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 \(G_1\square G_2\) of graphs \(G_1\) and \(G_2\) satisfying that \(G_1\) is traceable and \(G_2\) is connected with a path factor. Let \(P_n\) 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) \(P_n\square H\) is Hamiltonian; (ii) \(P_n\square H\) is 1-tough; and (iii) H has a path factor.

中文翻译：

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哈密​​顿量与笛卡尔积图的1韧性之间的关系

图的汉密尔顿性和韧性之间的关系是一个长期存在的研究问题。本文研究了图\（G_1 \）和\（G_2 \）的笛卡尔积图\（G_1 \ Square G_2 \）的哈密顿性，满足\（G_1 \）是可追溯的并且\（G_2 \）与a路径因素。让\（P_N \）是顺序路径Ñ和ħ是连接二分图。在满足n的特定条件下，我们证明以下三个语句是等价的：（i）\（P_n \ square H \）是哈密顿量；（ii）\（P_n \ square H \）是1-tough; 和（iii）H 有一个路径因素。