We provide precise asymptotic estimates for the number of several classes of labeled cubic planar graphs, and we analyze properties of such random graphs under the uniform distribution. This model was first analyzed by Bodirsky and coworkers. We revisit their work and obtain new results on the enumeration of cubic planar graphs and on random cubic planar graphs. In particular, we determine the exact probability of a random cubic planar graph being connected, and we show that the distribution of the number of triangles in random cubic planar graphs is asymptotically normal with linear expectation and variance. To the best of our knowledge, this is the first time one is able to determine the asymptotic distribution for the number of copies of a fixed graph containing a cycle in classes of random planar graphs arising from planar maps.

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关于随机三次平面图的进一步结果

我们提供了几类带标签的立方平面图的数量的精确渐近估计，并且我们分析了这种均匀分布下的随机图的性质。该模型首先由Bodirsky及其同事进行分析。我们重新审视他们的工作，并在立方平面图的枚举和随机立方平面图上获得新的结果。特别是，我们确定了连接随机三次平面图的确切概率，并且我们证明了随机三次平面图中三角形数目的分布是渐近正态的，具有线性期望和方差。就我们所知，这是第一次能够确定包含平面图的随机平面图类中包含循环的固定图的副本数的渐近分布。