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<p>We consider the number of crossings in a random embedding of a graph, <i>G</i>, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of <i>G</i>. Using Stein’s method and size-biased coupling, we establish an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. We also consider the case where <i>G</i> is a random graph and obtain a Kolmogorov bound between the distribution of crossings and a Gaussian mixture distribution. As applications, we obtain central limit theorems with convergence rates for the number of crossings in random embeddings of matchings, path graphs, cycle graphs, disjoint union of triangles, random <i>d</i>-regular graphs, and mixtures of random graphs.</p>

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Central Limit Theorem for Crossings in Randomly Embedded Graphs

  • 期刊论文

  • 发表日期：

    2025年6月20日

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  • Santiago Arenas-Velilla,
  • Octavio Arizmendi,
  • J. E. Paguyo

摘要

We consider the number of crossings in a random embedding of a graph, G, with vertices in convex position. We give explicit formulas for the mean and variance of the number of crossings as a function of various subgraph counts of G. Using Stein’s method and size-biased coupling, we establish an upper bound on the Kolmogorov distance between the distribution of the number of crossings and a standard normal random variable. We also consider the case where G is a random graph and obtain a Kolmogorov bound between the distribution of crossings and a Gaussian mixture distribution. As applications, we obtain central limit theorems with convergence rates for the number of crossings in random embeddings of matchings, path graphs, cycle graphs, disjoint union of triangles, random d-regular graphs, and mixtures of random graphs.