Closing the Random Graph Gap in Tuza's Conjecture Through the Online Triangle Packing Process

TL;DR

This paper proves Tuza's conjecture for Erdős-Rényi random graphs across all edge densities by analyzing an online triangle packing process with differential equations.

Contribution

It establishes the conjecture in random graphs for all ranges of edges, using a novel online process and differential equations analysis.

Findings

Tuza's conjecture holds in Erdős-Rényi graphs for all m.

The online triangle packing process effectively approximates the triangle packing number.

The differential equations method provides rigorous analysis of the process.

Abstract

A long-standing conjecture of Zsolt Tuza asserts that the triangle covering number $\tau(G)$ is at most twice the triangle packing number $\nu(G)$, where the triangle packing number $\nu(G)$ is the maximum size of a set of edge-disjoint triangles in $G$ and the triangle covering number $\tau(G)$ is the minimal size of a set of edges intersecting all triangles. In this paper, we prove that Tuza's conjecture holds in the Erd\H{o}s-R\'enyi random graph $G(n,m)$ for all range of $m$, closing the gap in what was previously known. (Recently, this result was also independently proved by Jeff Kahn and Jinyoung Park.) We employ a random greedy process called the online triangle packing process to produce a triangle packing in $G(n,m)$ and analyze this process by using the differential equations method.