SOLVED
Let $k\geq 3$ and $n$ be sufficiently large. Is it true that if $G$ is a graph with $n$ vertices and $2n-2$ edges such that every proper induced subgraph has minimum degree $\leq 2$ then $G$ must contain a copy of $C_k$?
In
[Er91] Erdős attributes this to himself and Hajnal, claiming they could prove it for $3\leq k\leq 6$, but it appears in an earlier paper of Erdős, Faudree, Gyárfás, and Schelp
[EFGS88], where they prove that such a graph on $n\geq 5$ vertices contains cycles of length $3$, $4$, and $5$, and a cycle of length at least $\lfloor \log_2n\rfloor$, and need not contain a cycle of length longer than $\sqrt{n}$.
Such graphs are called degree $3$ critical. This conjecture was disproved by Narins, Pokrovskiy, and Szabó [NPS17], who proved that there are arbitrarily large such graphs with no cycle of length $23$.
It remains open whether this question has an affirmative answer if we restrict to even $k$.
