The typical structure of sparse kr+1-free graphs

József Balogh, Robert Morris, Wojciech Samotij, Lutz Warnke

Research output: Contribution to journalArticlepeer-review

Abstract

Two central topics of study in combinatorics are the so-called evolution of random graphs, introduced by the seminal work of Erdős and Rényi, and the family of H-free graphs, that is, graphs which do not contain a subgraph isomorphic to a given (usually small) graph H. A widely studied problem that lies at the interface of these two areas is that of determining how the structure of a typical H-free graph with n vertices and m edges changes as m grows from 0 to ex(n, H). In this paper, we resolve this problem in the case when H is a clique, extending a classical result of Kolaitis, Prömel, and Rothschild. In particular, we prove that for every r ≥ 2 there is an explicit constant θγ such that, letting (Formula Precented) the following holds for every positive constant ε. If m ≥ (1 + ε)mγ then almost all Kγ+1-free n-vertex graphs with m edges are γ-partite, whereas if n ≪ m ≼ (1 + ε)mγ then almost all of them are not γ-partite.

Original languageEnglish (US)
Pages (from-to)6439-6485
Number of pages47
JournalTransactions of the American Mathematical Society
Volume368
Issue number9
DOIs
StatePublished - 2016

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

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