关于包含三条指定边的 Tutte 圈
On Tutte cycles containing three prescribed edges
AI总结：
本文证明了2-连通平面图中包含面圈三条指定边的Tutte圈存在性定理的定量版本，界定了删除Tutte圈后图的连通分支数量，并据此在本质4-连通平面图中找到了包含三条指定边的长圈。
摘要：
图 $G$ 中的圈 $C$ 被称为 Tutte 圈，如果从 $G$ 中删除 $C$ 后，每个连通分支在 $C$ 上最多有三个邻居。Tutte 圈在平面图 Hamilton 性的研究中起着重要作用。Thomas 和 Yu 以及独立地 Sanders 证明了在 2-连通平面图中存在包含一个面圈的三条指定边的 Tutte 圈。我们证明了这一结果的定量版本，界定了通过删除 Tutte 圈所得到的图的连通分支数量。作为推论，我们能在本质 4-连通平面图中找到长圈，且这些长圈也包含一个面圈的三条指定边。
英文摘要：
A cycle $C$ in a graph $G$ is called a Tutte cycle if, after deleting $C$ from $G$, each component has at most three neighbors on $C$. Tutte cycles play an important role in the study of Hamiltonicity of planar graphs. Thomas and Yu and independently Sanders proved the existence of Tutte cycles containining three specified edges of a facial cycle in a 2-connected plane graph. We prove a quantitative version of this result, bounding the number of components of the graph obtained by deleting a Tutte cycle. As a corollary, we can find long cycles in essentially 4-connected plane graphs that also contain three prescribed edges of a facial cycle.
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