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The number of K m,m -free graphs

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Abstract

A graph is called H-free if it contains no copy of H. Denote by f n (H) the number of (labeled) H-free graphs on n vertices. Erdős conjectured that f n (H) ≤ 2(1+o(1))ex(n,H). This was first shown to be true for cliques; then, Erdős, Frankl, and Rödl proved it for all graphs H with χ(H)≥3. For most bipartite H, the question is still wide open, and even the correct order of magnitude of log2 f n (H) is not known. We prove that f n (K m,m ) ≤ 2O(n 2−1/m) for every m, extending the result of Kleitman and Winston and answering a question of Erdős. This bound is asymptotically sharp for m∈{2,3}, and possibly for all other values of m, for which the order of ex(n,K m,m ) is conjectured to be Θ(n 2−1/m). Our method also yields a bound on the number of K m,m -free graphs with fixed order and size, extending the result of Füredi. Using this bound, we prove a relaxed version of a conjecture due to Haxell, Kohayakawa, and Łuczak and show that almost all K 3,3-free graphs of order n have more than 1/20·ex(n,K 3,3) edges.

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Authors and Affiliations

  1. Department of Mathematics, University of California, San Diego 9500 Gilman Drive, La Jolla, CA, 92093, USA

    József Balogh

  2. Department of Mathematics, University of Illinois, 1409 W. Green Street, Urbana, IL, 61801, USA

    József Balogh

  3. Department of Mathematics, University of Illinois, Urbana, IL, 61801, USA

    Wojciech Samotij

Authors
  1. József Balogh
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  2. Wojciech Samotij
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Correspondence to József Balogh.

Additional information

This material is based upon work supported by NSF CAREER Grant DMS-0745185, UIUC Campus Research Board Grant 09072, and OTKA Grant K76099.

Research supported in part by the Trijtzinsky Fellowship and the James D. Hogan Memorial Scholarship Fund.

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Balogh, J., Samotij, W. The number of K m,m -free graphs. Combinatorica 31, 131–150 (2011). https://doi.org/10.1007/s00493-011-2610-y

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