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Saturation in random graphs
Author(s) -
Korándi Dániel,
Sudakov Benny
Publication year - 2017
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20703
Subject(s) - combinatorics , random graph , random regular graph , mathematics , struct , graph , discrete mathematics , saturation (graph theory) , induced subgraph , line graph , pathwidth , computer science , vertex (graph theory) , programming language
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