arXiv - CS - Discrete Mathematics Pub Date : 2020-02-18 , DOI: arxiv-2002.07925
Fábio Botler; Cristina G. Fernandes; Juan Gutiérrez

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.

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