The structure of almost all graphs in a hereditary property

Noga Alon, József Balogh, Béla Bollobás, Robert Morris

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42 Scopus citations

Abstract

A hereditary property of graphs is a collection of graphs which is closed under taking induced subgraphs. The speed of P is the function n→|Pn|, where Pn denotes the graphs of order n in P. It was shown by Alekseev, and by Bollobás and Thomason, that if P is a hereditary property of graphs then. |Pn|=2(1-1/r+o(1))(n2), where r=r(P)j{cyrillic,ukrainian}N is the so-called 'colouring number' of P. However, their results tell us very little about the structure of a typical graph GεP. In this paper we describe the structure of almost every graph in a hereditary property of graphs, P. As a consequence, we derive essentially optimal bounds on the speed of P, improving the Alekseev-Bollobás-Thomason Theorem, and also generalising results of Balogh, Bollobás and Simonovits.

Original languageEnglish (US)
Pages (from-to)85-110
Number of pages26
JournalJournal of Combinatorial Theory. Series B
Volume101
Issue number2
DOIs
StatePublished - Mar 2011

All Science Journal Classification (ASJC) codes

  • Theoretical Computer Science
  • Discrete Mathematics and Combinatorics
  • Computational Theory and Mathematics

Keywords

  • Entropy
  • Hereditary property
  • Structure of graphs

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