The ratio of the numbers of odd and even cycles in outerplanar graphs

TL;DR

This paper studies the ratio of odd to even cycles in outerplanar graphs, showing divergence behavior, providing estimations, and exploring related problems in constrained graph classes.

Contribution

It offers new insights into cycle ratio behavior in outerplanar graphs and extends analysis to graphs with forbidden subgraphs and minors.

Findings

Ratio diverges to infinity as graph size increases

Provides sharp estimations for specific classes

Establishes a constant upper bound for some classes

Abstract

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.