Häggkvist–Jackson strengthening of Woodall's conjecture

Let $G$ be a 2-connected graph on $n$ vertices. Häggkvist–Jackson conjecture. If $G$ contains at least

$$\max\left\{2k-1,\frac{n+k}{2}+1\right\}$$

vertices of degree at least $k$, then $G$ has a cycle of length at least

$$\min\{n,2k\}.$$

The conjecture strengthens Woodall's circumference condition and is sharp for the two graph constructions described in the paper. Its resolution is not supplied here.