Triangle-free subgraphs of random graphs

TL;DR

This paper investigates the structure of triangle-free subgraphs in random graphs with high minimum degree, showing they are close to bipartite or r-partite, extending classical extremal graph theory results to random graphs.

Contribution

It establishes that high minimum degree triangle-free subgraphs in random graphs are structurally close to bipartite or r-partite graphs, generalizing classical results to the probabilistic setting.

Findings

Triangle-free subgraphs with high minimum degree are close to bipartite.

Such subgraphs are also close to r-partite for some r depending on epsilon.

Results are asymptotically optimal up to a constant factor.

Abstract

Recently there has been much interest in studying random graph analogues of well known classical results in extremal graph theory. Here we follow this trend and investigate the structure of triangle-free subgraphs of $G(n,p)$ with high minimum degree. We prove that asymptotically almost surely each triangle-free spanning subgraph of $G(n,p)$ with minimum degree at least $\big(\frac{2}{5} + o(1)\big)pn$ is $\mathcal O(p^{-1}n)$-close to bipartite, and each spanning triangle-free subgraph of $G(n,p)$ with minimum degree at least $(\frac{1}{3}+\varepsilon)pn$ is $\mathcal O(p^{-1}n)$-close to $r$-partite for some $r=r(\varepsilon)$. These are random graph analogues of a result by Andr\'asfai, Erd\H{o}s, and S\'os [Discrete Math. 8 (1974), 205-218], and a result by Thomassen [Combinatorica 22 (2002), 591--596]. We also show that our results are best possible up to a constant factor.