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 $G$.

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关于小树宽三角剖分和图的图扎猜想

Tuza (1981) 推测与图 $G$ 的每个三角形相交的最小边集的大小 $\tau(G)$ 最多是最大边集的大小 $\nu(G)$ 的两倍-$G$ 的不相交三角形。在本文中，我们展示了关于图扎猜想的三个结果。我们对树宽最多 $6$ 的图进行验证；我们证明 $\tau(G)\leq \frac{3}{2}\,\nu(G)$ 对于每个平面三角剖分 $G$ 与 $K_4$ 不同；并且 $\tau(G)\leq\frac{9}{5}\,\nu(G) + \frac{1}{5}$ 如果 $G$ 是树宽为 3 的最大图。我们的第一个结果加强了 Tuza 的结果，这意味着 $\tau(G) \leq 2\,\nu(G)$ 对于每个无 $K_8$ 的和弦图 $G$。