The structure of almost all graphs in a hereditary property

  • Authors:

  • Noga Alon;József Balogh;Béla Bollobás;Robert Morris

  • Affiliations:

  • Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel and IAS, Princeton, NJ 08540, USA;Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL 61801, USA and Department of Mathematics, University of California, San Diego, La Jolla, CA 92093, USA;Trinity College, Cambridge CB2 1TQ, England, United Kingdom and Department of Mathematical Sciences, The University of Memphis, Memphis, TN 38152, USA;IMPA, Estrada Dona Castorina 110, Jardim Botínico, Rio de Janeiro, RJ, Brazil

  • Venue:
  • Journal of Combinatorial Theory Series B
  • Year:
  • 2011

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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@?|P"n|, where P"n denotes the graphs of order n in P. It was shown by Alekseev, and by Bollobas and Thomason, that if P is a hereditary property of graphs then|P"n|=2^(^1^-^1^/^r^+^o^(^1^)^)^(^n^2^), where r=r(P)@?N is the so-called 'colouring number' of P. However, their results tell us very little about the structure of a typical graph G@?P. 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-Bollobas-Thomason Theorem, and also generalising results of Balogh, Bollobas and Simonovits.