Large triangle packings and Tuza's conjecture in sparse random graphs

TL;DR

This paper proves Tuza's conjecture for sparse and dense random graphs by analyzing a greedy algorithm for large triangle packings, establishing bounds on the number of edges intersecting all triangles.

Contribution

It demonstrates that Tuza's conjecture holds in certain regimes of the random graph G(n,m) by analyzing a specific greedy algorithm.

Findings

Tuza's conjecture holds for m ≤ 0.2403n^{3/2} in G(n,m)

Tuza's conjecture holds for m ≥ 2.1243n^{3/2} in G(n,m)

Analysis of a greedy algorithm enables these results.

Abstract

The triangle packing number $\nu(G)$ of a graph $G$ is the maximum size of a set of edge-disjoint triangles in $G$. Tuza conjectured that in any graph $G$ there exists a set of at most $2\nu(G)$ edges intersecting every triangle in $G$. We show that Tuza's conjecture holds in the random graph $G=G(n,m)$, when $m \le 0.2403n^{3/2}$ or $m\ge 2.1243n^{3/2}$. This is done by analyzing a greedy algorithm for finding large triangle packings in random graphs.