Thomassen proved that 4-connected planar graphs are Hamilton connected by showing that every 2-connected planar graph G contains a Tutte path P between any two given vertices, that is, every component of $G-P$ has at most three neighbors on P. In this paper, we prove a quantitative version of this result for circuit graphs, a natural class of planar graphs which includes all 3-connected planar graphs, by further controlling the number of components in $G-P$. We also give an application of this result by providing a best possible bound for the circumference of essentially 4-connected planar graphs.

中文翻译：

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电路图中的 Tutte 路径和长周期

Thomassen 通过证明每个 2 连通平面图G在任意两个给定顶点之间都包含一条 Tutte 路径P证明了 4 连通平面图是 Hamilton 连通的，即$G-磷$在P上最多有三个邻居。在本文中，我们通过进一步控制电路图中的组件数量，证明了电路图的这一结果的定量版本，电路图是一类自然平面图，包括所有 3 连通平面图。$G-磷$. 我们还通过为基本上 4 连通平面图的圆周提供最佳可能边界来应用此结果。