2009/02/03 by Asaf Shapira, Shapira, Asaf, Raphael Yuster +1
Mathematics · #05D99 #Combinatorics (math.CO) #FOS: Mathematics #math.CO #msc:05D99
paper · pdf · doi:10.48550/arxiv.0902.0580
arxiv created 2009/02/03 · arxiv updated 2009/12/01
The triangle removal lemma states that a simple graph with o(n3) triangles can be made triangle-free by removing o(n2) 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 n2+o(1) triangles that are still far from being triangle-free. On the other hand, we show that for some g(n)=ω(1), if a multi-graph (or weighted graph) has only g(n)n2 triangles then it must be close to being triangle-free. The proof relies on variants of the Ruzsa-Szemerédi theorem.