Weakly pancyclic vertices in dense nonbipartite graphs

Let $G$ be a graph of girth $g$ and circumference $c\mathrm{.}$ A vertex $v$ of $G$ is called weakly pancyclic if $v$ lies on an $\mathrm{\ell }$-cycle for every integer $\mathrm{\ell }$ with $g\le \mathrm{\ell }\le c\mathrm{.}$ We prove that if $G$ is a nonbipartite graph of order $n\ge 5$ and size at least $\lfloor (n-1{)}^2\mathrm{/}4\rfloor +2,$ then $G$ contains three weakly pancyclic vertices, with one exception. This strengthens a result of Brandt from 1997. We also pose a related problem.