Triangle packing and covering 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 (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.