  Tuza (1981) conjectured that the size $\tau(G)$ of a minimum set of edges
that intersects every triangle of a graph $G$ is at most twice the size
$\nu(G)$ of a maximum set of edge-disjoint triangles of $G$. In this paper we
present three results regarding Tuza's Conjecture. We verify it for graphs with
treewidth at most $6$; we show that $\tau(G)\leq \frac{3}{2}\,\nu(G)$ for every
planar triangulation $G$ different from $K_4$; and that
$\tau(G)\leq\frac{9}{5}\,\nu(G) + \frac{1}{5}$ if $G$ is a maximal graph with
treewidth 3. Our first result strengthens a result of Tuza, implying that
$\tau(G) \leq 2\,\nu(G)$ for every $K_8$-free chordal graph $G$.
