Multigraphs (only) satisfy a weak triangle removal lemma
Combinatorics
2009-02-04 v1
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.
Keywords
Cite
@article{arxiv.0902.0580,
title = {Multigraphs (only) satisfy a weak triangle removal lemma},
author = {Asaf Shapira and Raphael Yuster},
journal= {arXiv preprint arXiv:0902.0580},
year = {2009}
}