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

Abstract Tuza (1981) conjectured that the size  τ ( G ) of a minimum set of edges that intersects every triangle of a graph  G is at most twice the size  ν ( 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 τ ( G ) ≤ 3 2 ν ( G ) for every planar triangulation G different from K 4 ; and that τ ( G ) ≤ 9 5 ν ( G ) + 1 5 if G is a maximal graph with treewidth 3. Our first result strengthens a result of Tuza, implying that τ ( G ) ≤ 2 ν ( G ) for every K 8 -free chordal graph G .