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

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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${x_{n+1} = 1 + x_n^2, \, x_0=1}$$\end{document}. The same result holds for the larger class of series–parallel graphs.