In 1982, Tuza 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 $G$. This conjecture was proved for several graph classes. In this paper, we present three results regarding Tuza's Conjecture for dense graphs. By using a probabilistic argument, Tuza proved its conjecture for graphs on $n$ vertices with minimum degree at least $\frac{7n}{8}$. We extend this technique to show that Tuza's conjecture is valid for split graphs with minimum degree at least $\frac{3n}{5}$; and that $\tau(G) < \frac{28}{15}\nu(G)$ for every tripartite graph with minimum degree more than $\frac{33n}{56}$. Finally, we show that $\tau(G)\leq \frac{3}{2}\nu(G)$ when $G$ is a complete 4-partite graph. Moreover, this bound is tight.

翻译：1982年，图扎猜想：对于任意图$G$，与所有三角形相交的最小边集的大小$\tau(G)$至多为最大边不交三角形集大小$\nu(G)$的两倍。该猜想已在若干图类中得到证明。本文针对密集图，给出了关于图扎猜想的三个结果。图扎通过概率论证证明了最小度至少为$\frac{7n}{8}$的$n$顶点图满足该猜想。我们扩展了这一技术，证明最小度至少为$\frac{3n}{5}$的分裂图满足图扎猜想；且对于最小度大于$\frac{33n}{56}$的三部图，有$\tau(G) < \frac{28}{15}\nu(G)$。最后，我们证明当$G$为完全四部图时，$\tau(G)\leq \frac{3}{2}\nu(G)$，且该界是紧的。