On the Maximum Number of Cycles in Outerplanar and Series–Parallel Graphs

Abstract

Let c(n) be the maximum number of cycles in an outerplanar graph with n vertices. We show that lim c(n)1/n exists and equals β = 1.502837 . . ., where β is a constant related to the recurrence $${x_{n+1} = 1 + x_n^2, \, x_0=1}$$. The same result holds for the larger class of series–parallel graphs.