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<p>An planar graph is a graph that can be drawn on a plane without intersecting edges. A pentacyclic graph is a connected graph with <i>n</i> vertices and <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(n + 4\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>+</mo> <mn>4</mn> </mrow> </math></EquationSource> </InlineEquation> edges. We obtain an explicit formula for the number of labeled nonplanar pentacyclic blocks with a given number of vertices and found the corresponding asymptotics for the number of such graphs with a large number of vertices. We prove that under the uniform probability distribution, the probability that the labeled pentacyclic block is a nonplanar graph is asymptotically equal to&#xa0;80/539.</p>

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ENUMERATION OF LABELED NONPLANAR PENTACYCLIC BLOCKS

  • 期刊论文

  • 发表日期：

    2026年2月1日

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  • V. A. Voblyi

摘要

An planar graph is a graph that can be drawn on a plane without intersecting edges. A pentacyclic graph is a connected graph with n vertices and \(n + 4\) n + 4 edges. We obtain an explicit formula for the number of labeled nonplanar pentacyclic blocks with a given number of vertices and found the corresponding asymptotics for the number of such graphs with a large number of vertices. We prove that under the uniform probability distribution, the probability that the labeled pentacyclic block is a nonplanar graph is asymptotically equal to 80/539.