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Avoiding small subgraphs in Achlioptas processes
Author(s) -
Krivelevich Michael,
Loh PoShen,
Sudakov Benny
Publication year - 2009
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.20254
Subject(s) - combinatorics , random graph , mathematics , graph , bipartite graph , giant component , random regular graph , discrete mathematics , line graph , pathwidth
For a fixed integer r , consider the following random process. At each round, one is presented with r random edges from the edge set of the complete graph on n vertices, and is asked to choose one of them. The selected edges are collected into a graph, which thus grows at the rate of one edge per round. This is a natural generalization of what is known in the literature as an Achlioptas process (the original version has r = 2), which has been studied by many researchers, mainly in the context of delaying or accelerating the appearance of the giant component. In this article, we investigate the small subgraph problem for Achlioptas processes. That is, given a fixed graph H , we study whether there is an online algorithm that substantially delays or accelerates a typical appearance of H , compared to its threshold of appearance in the random graph G ( n, M ). It is easy to see that one cannot accelerate the appearance of any fixed graph by more than the constant factor r , so we concentrate on the task of avoiding H . We determine thresholds for the avoidance of all cycles C t , cliques K t , and complete bipartite graphs K t,t , in every Achlioptas process with parameter r ge 2. © 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009