A Counter Example to a Conjecture of Gallai

Let \(G\) be a graph and \(t(G)\) be the number of triangles in \(G\). Define \(\mathcal G_n\) to be the set of all graphs on \(n\) vertices that do not contain a wheel and \(t_n = \max\{t(G) : G \in \mathcal G_n\}\).
T. Gallai conjectured that \(t_n \leq \lfloor\frac{n^2}{8}\rfloor\). In this note we describe a graph on \(n\) vertices that contains no wheel and has at least \(\frac{n^2+n}{8}-3\) triangles.