  In this paper, we investigate the ratio of the numbers of odd and even cycles
in outerplanar graphs. We verify that the ratio generally diverges to infinity
as the order of a graph diverges to infinity. We also give sharp estimations of
the ratio for several classes of outerplanar graphs, and obtain a constant
upper bound of the ratio for some of them. Furthermore, we consider similar
problems in graphs with some pairs of forbidden subgraphs/minors, and propose a
challenging problem concerning claw-free graphs.
