Many disjoint triangles in co-triangle-free graphs

TL;DR

This paper proves that graphs with triangle-free complements contain nearly the maximum possible number of edge-disjoint triangles, and shows that extremal graphs are close to bipartite, advancing understanding of Erdős's conjecture.

Contribution

It establishes tight bounds on the number of edge-disjoint triangles in graphs with triangle-free complements and provides a stability result characterizing extremal structures.

Findings

Graphs with triangle-free complements contain about n^2/12 edge-disjoint triangles.

Extremal graphs are structurally close to bipartite graphs.

The results answer a question of Alon and Linial and progress on Erdős's conjecture.

Abstract

We prove that any $n$-vertex graph whose complement is triangle-free contains $n^2/12-o(n^2)$ edge-disjoint triangles. This is tight for the disjoint union of two cliques of order $n/2$. We also prove a corresponding stability theorem, that all large graphs attaining the above bound are close to being bipartite. Our results answer a question of Alon and Linial, and make progress on a conjecture of Erd\H{o}s.