The number of Ks,t-free graphs

József Balogh; Wojciech Samotij

doi:10.1112/jlms/jdq086

The number of K_s,t-free graphs

József Balogh^*, Wojciech Samotij

^*Corresponding author for this work

University of Illinois at Urbana-Champaign

Research output: Contribution to journal › Article › peer-review

30 Scopus citations

Abstract

Denote by f_n (H) the number of (labeled) H-free graphs on a fixed vertex set of size n. Erdos conjectured that, whenever H contains a cycle, f_n(H) = 2^{(1+0(1))ex(n,h)}, yet it is still open for every bipartite graph, and even the order of magnitude of log₂ f _n (H) was known only for C₄, C₆, and K _3,3. We show that, for all s and t satisfying 2≤s≤t, f _n(K_s,t=2^O(n2-1/s), which is asymptotically sharp for those values of s and t for which the order of magnitude of the Turán number ex(n, K_s,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c such that almost all K_2,t-free graphs of order n have at least 1/12 · ex(n, K_2,t) and at most (1 - c) ex(n, K_2,t) edges. Moreover, our results have some interesting applications to the study of some Ramsey- and Turán-type problems.

Original language	English
Pages (from-to)	368-388
Number of pages	21
Journal	Journal of the London Mathematical Society
Volume	83
Issue number	2
DOIs	https://doi.org/10.1112/jlms/jdq086
State	Published - Apr 2011
Externally published	Yes

Funding

Funders	Funder number
James D. Hogan Memorial Scholarship Fund
UIUC Campus Research Board	08086, 09072
National Science Foundation	DMS-0745185, DMS-0600303
Hungarian Scientific Research Fund	K76099

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10.1112/jlms/jdq086

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title = "The number of Ks,t-free graphs",

abstract = "Denote by fn (H) the number of (labeled) H-free graphs on a fixed vertex set of size n. Erdos conjectured that, whenever H contains a cycle, fn(H) = 2(1+0(1))ex(n,h), yet it is still open for every bipartite graph, and even the order of magnitude of log2 f n (H) was known only for C4, C6, and K 3,3. We show that, for all s and t satisfying 2≤s≤t, f n(Ks,t=2O(n2-1/s), which is asymptotically sharp for those values of s and t for which the order of magnitude of the Tur{\'a}n number ex(n, Ks,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c such that almost all K2,t-free graphs of order n have at least 1/12 · ex(n, K2,t) and at most (1 - c) ex(n, K2,t) edges. Moreover, our results have some interesting applications to the study of some Ramsey- and Tur{\'a}n-type problems.",

author = "J{\'o}zsef Balogh and Wojciech Samotij",

note = "Funding Information: The first author was supported by NSF CAREER Grant DMS-0745185 and DMS-0600303, UIUC Campus Research Board Grants 09072 and 08086, and OTKA Grant K76099. The second author was supported in part by Trijtzinsky Fellowship and James D. Hogan Memorial Scholarship Fund.",

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N2 - Denote by fn (H) the number of (labeled) H-free graphs on a fixed vertex set of size n. Erdos conjectured that, whenever H contains a cycle, fn(H) = 2(1+0(1))ex(n,h), yet it is still open for every bipartite graph, and even the order of magnitude of log2 f n (H) was known only for C4, C6, and K 3,3. We show that, for all s and t satisfying 2≤s≤t, f n(Ks,t=2O(n2-1/s), which is asymptotically sharp for those values of s and t for which the order of magnitude of the Turán number ex(n, Ks,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c such that almost all K2,t-free graphs of order n have at least 1/12 · ex(n, K2,t) and at most (1 - c) ex(n, K2,t) edges. Moreover, our results have some interesting applications to the study of some Ramsey- and Turán-type problems.

AB - Denote by fn (H) the number of (labeled) H-free graphs on a fixed vertex set of size n. Erdos conjectured that, whenever H contains a cycle, fn(H) = 2(1+0(1))ex(n,h), yet it is still open for every bipartite graph, and even the order of magnitude of log2 f n (H) was known only for C4, C6, and K 3,3. We show that, for all s and t satisfying 2≤s≤t, f n(Ks,t=2O(n2-1/s), which is asymptotically sharp for those values of s and t for which the order of magnitude of the Turán number ex(n, Ks,t) is known. Our methods allow us to prove, among other things, that there is a positive constant c such that almost all K2,t-free graphs of order n have at least 1/12 · ex(n, K2,t) and at most (1 - c) ex(n, K2,t) edges. Moreover, our results have some interesting applications to the study of some Ramsey- and Turán-type problems.

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The number of K_s,t-free graphs

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