Before revision:  However, the  best known upper bound is given by an infinite family of such graphs in which every graph G on n vertices has no cycle longer than\left \lfloor \frac{2}{3}(n+4) \right \rfloor ;
After revision:  However, the best known upper bound is given by an infinite family of such graphs in which no graph G contains a cycle that is longer than\left \lfloor \frac{2}{3}(n+4) \right \rfloor ;