Tuza's Conjecture for Graphs of Maximum Degree at Most 10

For a finite simple graph $G$, let $\nu (G)$ be the maximum number of pairwise edge disjoint triangles and $\tau (G)$ the minimum number of edges meeting every triangle. Tuza conjectured in 1981 that every such $G$ satisfies $\tau (G)\le 2\nu (G)\mathrm{.}$ We prove the conjecture for the class of graphs with $\mathrm{\Delta }\le 10.$