Saturation in random graphs. Issue 1 (4th November 2016)

• Record Type: Journal Article
• Authors: Korándi, Dániel; Sudakov, Benny
• Is Part Of: Random structures & algorithms. Volume 51:Issue 1(2017)
• Page Start: 169
• Page End: 181
• Publication Date: 2016-11-04
• Subjects: saturation -- graph bootstrap percolation -- random graphs; Random graphs -- Periodicals; Mathematical analysis -- Periodicals; 519
• DOI: 10.1002/rsa.20703
• Languages: English
• ISSNs: 1042-9832

Abstract

ABSTRACT: A graph H is K s ‐saturated if it is a maximal K s ‐free graph, i.e., H contains no clique on s vertices, but the addition of any missing edge creates one. The minimum number of edges in a K s ‐saturated graph was determined over 50 years ago by Zykov and independently by Erdős, Hajnal and Moon. In this paper, we study the random analog of this problem: minimizing the number of edges in a maximal K s ‐free subgraph of the Erdős‐Rényi random graph G ( n, p ). We give asymptotically tight estimates on this minimum, and also provide exact bounds for the related notion of weak saturation in random graphs. Our results reveal some surprising behavior of these parameters. © 2016 Wiley Periodicals, Inc. Random Struct. Alg., 51, 169–181, 2017