Tuza's Conjecture for random graphs

TL;DR

This paper proves that Tuza's conjecture holds asymptotically almost surely for random graphs, showing that the minimum triangle cover size is at most twice the maximum set of edge-disjoint triangles with high probability.

Contribution

The paper establishes that Tuza's conjecture is asymptotically true for Erdős–Rényi random graphs, resolving a recent open question.

Findings

Tuza's conjecture holds asymptotically almost surely for $G_{n,p}$.

The probability that a random graph satisfies Tuza's conjecture approaches 1 as n grows.

The result applies for any edge probability p(n).

Abstract

A celebrated conjecture of Zs. Tuza says that in any (finite) graph, the minimum size of a cover of triangles by edges is at most twice the maximum size of a set of edge-disjoint triangles. Resolving a recent question of Bennett, Dudek, and Zerbib, we show that this is true for random graphs; more precisely: \[ \mbox{for any $p=p(n)$, $\mathbb P(\mbox{$G_{n,p}$ satisfies Tuza's Conjecture})\rightarrow 1 $ (as $n\rightarrow\infty$).} \]