Triangle-Free Subgraphs of Random Graphs

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 (2/5 + o(1)) pn is O(p(-1)n)-close to bipartite, and each spanning triangle-free subgraph of G(n, p) with minimum degree at least (1/3 + epsilon) pn is O(p(-1)n)-close to r-partite for some r = r(e). These are random graph analogues of a result by Andrasfai, Erdos and Sos (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.