On Tuza's Conjecture for Triangulations and Graphs with Small Treewidth

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Tuza (1981) conjectured that the cardinality τ(G) of a minimum set of edges that intersects every triangle of a graph G is at most twice the cardinality ν(G) of a maximum set of edge-disjoint triangles of G. I this paper we present three results regarding Tuza's Conjecture. We verify it for graphs with treewidth at most 6; and we show that τ(G)≤32ν(G) for every planar triangulation G different from K4; and that τ(G)≤95ν(G)+15 if G is a maximal graph with treewidth 3.