Multigraphs (only) satisfy a weak triangle removal lemma
The electronic journal of combinatorics, Tome 16 (2009) no. 1
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. 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 slowly growing function $g(n)=\omega(1)$, if a multi-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-Szemerédi theorem.
@article{10_37236_249,
author = {Asaf Shapira and Raphael Yuster},
title = {Multigraphs (only) satisfy a weak triangle removal lemma},
journal = {The electronic journal of combinatorics},
year = {2009},
volume = {16},
number = {1},
doi = {10.37236/249},
zbl = {1182.05067},
url = {http://geodesic.mathdoc.fr/articles/10.37236/249/}
}
Asaf Shapira; Raphael Yuster. Multigraphs (only) satisfy a weak triangle removal lemma. The electronic journal of combinatorics, Tome 16 (2009) no. 1. doi: 10.37236/249
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