Extremal Subgraphs of Random Graphs: an Extended Version

TL;DR

This paper proves that for sufficiently dense random graphs, all maximum triangle-free subgraphs are bipartite, and introduces a new method showing the near-uniqueness of maximum cuts in large graphs.

Contribution

It establishes a threshold for random graphs where maximum triangle-free subgraphs are bipartite and introduces a novel approach to analyze maximum cuts.

Findings

Maximum triangle-free subgraphs are bipartite for p ≥ n^{-c}.

Maximum cuts in large graphs are nearly unique, differing by small vertex moves.

Provides a new tool for analyzing maximum cuts in graphs with many edges.

Abstract

We prove that there is a constant $c >0$, such that whenever $p \ge n^{-c}$, with probability tending to 1 when $n$ goes to infinity, every maximum triangle-free subgraph of the random graph $G_{n,p}$ is bipartite. This answers a question of Babai, Simonovits and Spencer (Journal of Graph Theory, 1990). The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with $M$ edges, where $M >> n$, is ``nearly unique''. More precisely, given a maximum cut $C$ of $G_{n,M}$, we can obtain all maximum cuts by moving at most $O(\sqrt{n^3/M})$ vertices between the parts of $C$.