The odd–even cycle ratio conjecture for 2-connected claw-free graphs

Let $G$ be a finite 2-connected graph with no subgraph isomorphic to $K_{1,3}$. Write $c_o(G)$ and $c_e(G)$ for the numbers of odd and even cycles of $G$, respectively.

Odd–even cycle ratio conjecture. Both ratios

$$\frac{c_o(G)}{c_e(G)}$$

and

$$\frac{c_e(G)}{c_o(G)}$$

are bounded by a constant unless $G$ is a cycle.

This conjecture proposes that, apart from cycles, 2-connected $K_{1,3}$-free graphs have comparable numbers of odd and even cycles. The analogous assertion is known under additional hypotheses, but the general claim remains open; non-2-connected $K_{1,3}$-free graphs can have an unbounded ratio.