I'm trying to prove the following result in order to solve a different problem but I'm stuck; however I'm not sure if it is true, so I'll pose it as a question;
Suppose we have a triangle-free Hamiltonian graph $G$ and we know that $G$ contains a large induced cycle of even length. Moreover, suppose that $G$ contains a large number number of cycles that have consecutive lengths (where the two "large"'s are both of the same order if you think it's relevant). Is it true that $G$ must necessarily contains a large induced cycle of odd length?
Thank you in advance for any ideas!