Let be a graph chosen uniformly at random from the class of all planar graphs on vertex set with edges. We study the cycle and block structure of when . More precisely, we determine the asymptotic order of the length of the longest and shortest cycle in in the critical range when . In addition, we describe the block structure of in the weakly supercritical regime when .

中文翻译：

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随机平面图中的最长和最短循环

让是从所有平面图的类中均匀随机选择的一个图，该图是带有边的顶点集。我们研究when的循环和块结构。更准确地说，我们确定了临界范围内最长和最短循环长度的渐近顺序当。此外，我们描述了当 时的弱超临界状态中的块结构。