再论块的平方中的Hamilton主题：DT-图的情况
Revisiting the Hamiltonian Theme in the Square of a Block: The Case of DT-Graphs
摘要：
图G的平方，记为G^2，是由G通过用边连接任意两个有公共邻居的不相邻顶点而得到的图。称图G具有F_k性质，如果对任意k个不同顶点的集合{x_1，x_2，.，x_k}，则G ^2中存在一条从x_1到x_2的Hamilton路，其中包含G的k-2条不同边，形式为x_iz_i，i = 3，.，K.许多年前就证明了2-连通图具有F_3性质。在本文的第一部分中，我们推广了这个结果，证明了每个2-连通DT-图都具有F_4性质（定理2），并在第二部分中证明了这个推广对任意的2-连通图都成立，并且存在对任意自然数k >= 5不具有F_k性质的2-连通图。总之，这肯定地回答了以前提出的一个问题。
英文摘要：
The square of a graph G, denoted G^2, is the graph obtained from G by joining by an edge any two nonadjacent vertices which have a common neighbor. A graph G is said to have the F_k property if for any set of k distinct vertices {x_1, x_2, ..., x_k} in G, there is a hamiltonian path from x_1 to x_2 in G^2 containing k-2 distinct edges of G of the form x_iz_i, i = 3, ..., k. It was proved many years ago that every 2-connected graph has the F_3 property. In the first part of this work, we extend this result by proving that every 2-connected DT-graph has the F_4 property (Theorem 2) and will show in the second part that this generalization holds for arbitrary 2-connected graphs, and that there exist 2-connected graphs which do not have the F_k property for any natural number k >= 5. Altogether, this answers a problem raised before in the affirmative.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
