Triangle-Free Subgraphs of Random Graphs

Author(s)

Allen, Peter ^[1] ; Bottcher, Julia ^[1] ; Kohayakawa, Yoshiharu ^[2] ; Roberts, Barnaby ^[1]

Affiliation

^[1] London Sch Econ & Polit Sci, Dept Math, Houghton St, London WC2A 2AE - England

^[2] Univ Sao Paulo, Inst Matemat & Estat, Rua Matao 1010, BR-05508090 Sao Paulo - Brazil

• Document type: Journal article
• Source: COMBINATORICS PROBABILITY & COMPUTING; v. 27, n. 2, p. 141-161, MAR 2018.
• Web of Science Citations: 0

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 (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. (AU)

• FAPESP's process: 13/03447-6 - Combinatorial structures, optimization, and algorithms in theoretical Computer Science
• Grantee: Carlos Eduardo Ferreira
• Support Opportunities: Research Projects - Thematic Grants
• FAPESP's process: 13/07699-0 - Research, Innovation and Dissemination Center for Neuromathematics - NeuroMat
• Grantee: Oswaldo Baffa Filho
• Support Opportunities: Research Grants - Research, Innovation and Dissemination Centers - RIDC