Woodall's circumference conjecture

Let $G$) be a 2-connected graph on $n$ vertices, and let $c(G)$ denote the length of a longest cycle in $G$. Woodall's conjecture. If there are at least $\frac{n}{2}+k$ vertices of degree at least $k$, then

$$c(G)\geq 2k.$$

This conjecture is proved in the paper as an application of the strengthened Erdős–Gallai theorem.