Two counterexamples to a conjecture about even cycles

A conjecture of Verstraëte states that for any fixed $\mathrm{\ell }<k$ there exists a positive constant $c$ such that any $C_{2k}$-free graph $G$ contains a $C_{2\mathrm{\ell }}$-free subgraph with at least $c\mathrm{\mid }E(G)\mathrm{\mid }$ edges. For $\mathrm{\ell }=2$, this conjecture was verified by Kühn and Osthus in 2004. We identify two counterexamples to this conjecture for $\mathrm{\ell }=4$ and $k=5$: the first comes from a recent construction of a dense $C_{10}$-free subgraph of the hypercube and the second from Wenger’s construction for extremal $C_{10}$-free graphs.