Multigraphs (only) satisfy a weak triangle removal lemma

TL;DR

This paper demonstrates that the triangle removal lemma does not extend to multigraphs, but establishes a weak form of the lemma for multigraphs under certain conditions, using variants of the Ruzsa-Szemerédi theorem.

Contribution

It shows the limitations of extending the triangle removal lemma to multigraphs and introduces a weak triangle removal lemma for multigraphs with many triangles.

Findings

Multigraphs can have many triangles yet be far from triangle-free.

A weak triangle removal lemma holds for multigraphs with sufficiently many triangles.

The proof uses variants of the Ruzsa-Szemerédi theorem.

Abstract

The triangle removal lemma states that a simple graph with o(n^3) triangles can be made triangle-free by removing o(n^2) edges. It is natural to ask if this widely used result can be extended to multi-graphs (or equivalently, weighted graphs). In this short paper we rule out the possibility of such an extension by showing that there are multi-graphs with only n^{2+o(1)} triangles that are still far from being triangle-free. On the other hand, we show that for some g(n)=\omega(1), if a multi-graph (or weighted graph) has only g(n)n^2 triangles then it must be close to being triangle-free. The proof relies on variants of the Ruzsa-Szemer\'edi theorem.