Triangle-free subgraphs of random graphs

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.