On Tuza's conjecture for triangulations and graphs with small treewidth

Abstract

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 $\tau (G)\le \frac{3}{2}\nu (G)$ for every planar triangulation $G$ different from $K_4$; and that $\tau (G)\le \frac{9}{5}\nu (G)+\frac{1}{5}$ if $G$ is a maximal graph with treewidth 3. Our first result strengthens a result of Tuza, implying that $\tau (G)\le 2\nu (G)$ for every $K_8$-free chordal graph $G$.