Given a simple graph \(G=(V,E)\), a subset of E is called a triangle cover if it intersects each triangle of G. Let \(\nu _t(G)\) and \(\tau _t(G)\) denote the maximum number of pairwise edge-disjoint triangles in G and the minimum cardinality of a triangle cover of G, respectively. Tuza (in: Finite and infinite sets, proceedings of Colloquia Mathematica Societatis, Janos Bolyai, p 888, 1981) conjectured in 1981 that \(\tau _t(G)/\nu _t(G)\le 2\) holds for every graph G. In this paper, we consider Tuza’s Conjecture on dense random graphs. Under \(\mathcal {G}(n,p)\) model with a constant p, we prove that the ratio of \(\tau _t(G)\) and \(\nu _t(G)\) has the upper bound close to 1.5 with high probability. Furthermore, the ratio 1.5 is nearly the best result when \(p\ge 0.791\). In some sense, on dense random graphs, these conclusions verify Tuza’s Conjecture.

给定一个简单图\(G=(V,E)\) ，如果E的子集与G的每个三角形相交，则称为三角形覆盖。令\(\nu _t(G)\)和\(\tau _t(G)\)分别表示G中成对边不相交三角形的最大数量和G的三角形覆盖的最小基数。Tuza（in：Finite and Infinite Sets，Colloquia Mathematica Societatis，Janos Bolyai，第 888 页，1981 年）在 1981 年推测\(\tau _t(G)/\nu _t(G)\le 2\)对每个图G。在本文中，我们考虑了 Tuza 关于稠密随机图的猜想。在\(\mathcal {G}(n,p)\)下在具有常数p的模型中，我们证明了\(\tau _t(G)\)和\(\nu _t(G)\)的比率的上限接近 1.5 的概率很高。此外，当\(p\ge 0.791\)时，比率 1.5 几乎是最佳结果。在某种意义上，在稠密随机图上，这些结论验证了图萨猜想。