Packing and Covering Triangles in Dense Random Graphs

Abstract

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 [25] 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. We prove that under \(\mathcal {G}(n,p)\) model with \(p=\varOmega (1)\), for any \(0<\epsilon <1\), \(\tau _t(G)\le 1.5(1+\epsilon )\nu _t(G)\) holds with high probability, and under \(\mathcal {G}(n,m)\) model with \(m=\varOmega (n^2)\), for any \(0<\epsilon <1\), \(\tau _t(G)\le 1.5(1+\epsilon )\nu _t(G)\) holds with high probability. In some sense, on dense random graphs, these conclusions verify Tuza’s Conjecture.