Max Cuts in Triangle-free Graphs

TL;DR

This paper advances the understanding of Erd ext{o}s's conjecture by extending the edge density ranges for which it holds and improving the bounds on edges to remove for bipartization of triangle-free graphs.

Contribution

It extends the edge density ranges where Erd ext{o}s's conjecture is verified and improves the upper bound on edges to remove for bipartization.

Findings

Proves the conjecture for graphs with edge density up to 0.2486

Proves the conjecture for graphs with edge density at least 0.3197

Improves the maximum edges to remove to bipartite from n^2/18 to n^2/23.5

Abstract

A well-known conjecture by Erd\H{o}s states that every triangle-free graph on $n$ vertices can be made bipartite by removing at most $n^2/25$ edges. This conjecture was known for graphs with edge density at least $0.4$ and edge density at most $0.172$. Here, we will extend the edge density for which this conjecture is true; we prove the conjecture for graphs with edge density at most $0.2486$ and for graphs with edge density at least $0.3197$. Further, we prove that every triangle-free graph can be made bipartite by removing at most $n^2/23.5$ edges improving the previously best bound of $n^2/18$.