The typical structure of graphs without given excluded subgraphs

  • Authors:

  • József Balogh;Béla Bollobás;Miklós Simonovits

  • Affiliations:

  • Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801;Department of Pure Mathematics and Mathematical Statistics, Trinity College, Cambridge CB2 1TQ, UK and Department of Mathematical Sciences, University of Memphis, Memphis TN 38152;Alfred Rényi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, Hungary

  • Venue:
  • Random Structures & Algorithms
  • Year:
  • 2009

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Abstract

Let $ {\cal L}$ be a finite family of graphs. We describe the typical structure of $ {\cal L}$-free graphs, improving our earlier results (Balogh et al., J Combinat Theory Ser B 91 (2004), 1–24) on the Erdős–Frankl–Rödl theorem (Erdős et al., Graphs Combinat 2 (1986), 113–121), by proving our earlier conjecture that, for $ p = p ({\cal L}) = {\rm min}_L \in {\cal L} \chi (L) - 1 $, the structure of almost all $ {\cal L}$-free graphs is very similar to that of a random subgraph of the Turán graph Tn,p. The “similarity” is measured in terms of graph theoretical parameters of $ {\cal L}$.© 2008 Wiley Periodicals, Inc. Random Struct. Alg., 2009