An asymptotics for the number of labelled planar tetracyclic and pentacyclic graphs

Abstract

A connected graph with a cyclomatic number k is said to be a k-cyclic graph. We obtain the formula for the number of labelled non-planar pentacyclic graphs with a given number of vertices, and find the asymptotics of the number of labelled connected planar tetracyclic and pentacyclic graphs with n vertices as n → ∞. We prove that under a uniform probability distribution on the set of graphs under consideration, the probability that the labelled tetracyclic graph is planar is asymptotically equal to 1089/1105, and the probability that the labeled pentacyclic graph is planar is asymptotically equal to 1591/1675.