The square of a graph is obtained by adding an edge between each pair of vertices that are of distance 2 in the original graph. If a graph G has a hamiltonian path (or cycle) square, then the square of the hamiltonian path (or cycle) is called a hamiltonian path (or cycle) square of G. Chen and Shan (Graphs Comb 31:2113–2124, 2015) characterized all forbidden pairs for a 4-connected graph to have a hamiltonian cycle square. In this paper, we completely characterize pairs of connected forbidden subgraphs for a k-connected (\(k\in \{1,2,3,4\}\)) graph with maximal degree at least 4 to have a hamiltonian path square. Note that the maximal degree condition is necessary for graphs of order at least 5 to have a hamiltonian path square.

通过在原始图中距离为2的每对顶点之间添加一条边来获得图的平方。如果一个图ģ具有汉弥尔顿路径（或周期）的正方形，则汉弥尔顿路径（或周期）的平方被称为汉弥尔顿路径（或周期）的方形的ģ。Chen和Shan（Graphs Comb 31：2113–2124，2015）描绘了一个四连通图的所有禁止对都具有哈密顿循环平方。在本文中，我们完全刻画了k个连通（\（k \ in \ {1,2,3,4 \} \））最大度数至少为4的图形具有哈密顿路径平方。注意，最大阶数条件对于至少5阶的图形具有哈密顿路径平方是必要的。