Almost all C4-free graphs have fewer than (1 - ε)ex(n, C4) edges

József Balogh; Wojciech Samotij

doi:10.1137/09074989X

Almost all C₄-free graphs have fewer than (1 - ε)ex(n, C₄) edges

József Balogh^*
, Wojciech Samotij

^*Corresponding author for this work

University of California at San Diego
University of Illinois at Urbana-Champaign

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

5 Scopus citations

Abstract

A graph is called H-free if it contains no copy o f H. Let ex(n, H) denote the Turán number for H, i.e., the maximum number of edges that an n-vertex H-free graph may have. An old result of Kleitman and Winston states that there are 2^o(ex(n,C4)) C₄-free graphs on n vertices. Füredi showed that almost all C₄-free graphs of order n have at least cex(n, C₄) edges for some positive constant c. We prove that there is a positive constant e such that almost all C ₄-free graphs have at most (1 - ε) ex(n, C₄) edges. This resolves a conjecture of Balogh, Bollobás, and Simonovits for the 4-cycle.

Original language	English
Pages (from-to)	1011-1018
Number of pages	8
Journal	SIAM Journal on Discrete Mathematics
Volume	24
Issue number	3
DOIs	https://doi.org/10.1137/09074989X
State	Published - 2010
Externally published	Yes

Keywords

Asymptotic graph enumeration
Asymptotic graph structure
C-free
Extremal graphs
Turán's problem

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10.1137/09074989X

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title = "Almost all C4-free graphs have fewer than (1 - ε)ex(n, C4) edges",

abstract = "A graph is called H-free if it contains no copy o f H. Let ex(n, H) denote the Tur{\'a}n number for H, i.e., the maximum number of edges that an n-vertex H-free graph may have. An old result of Kleitman and Winston states that there are 2o(ex(n,C4)) C4-free graphs on n vertices. F{\"u}redi showed that almost all C4-free graphs of order n have at least cex(n, C4) edges for some positive constant c. We prove that there is a positive constant e such that almost all C 4-free graphs have at most (1 - ε) ex(n, C4) edges. This resolves a conjecture of Balogh, Bollob{\'a}s, and Simonovits for the 4-cycle.",

keywords = "Asymptotic graph enumeration, Asymptotic graph structure, C-free, Extremal graphs, Tur{\'a}n's problem",

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doi = "10.1137/09074989X",

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T1 - Almost all C4-free graphs have fewer than (1 - ε)ex(n, C4) edges

AU - Balogh, József

AU - Samotij, Wojciech

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N2 - A graph is called H-free if it contains no copy o f H. Let ex(n, H) denote the Turán number for H, i.e., the maximum number of edges that an n-vertex H-free graph may have. An old result of Kleitman and Winston states that there are 2o(ex(n,C4)) C4-free graphs on n vertices. Füredi showed that almost all C4-free graphs of order n have at least cex(n, C4) edges for some positive constant c. We prove that there is a positive constant e such that almost all C 4-free graphs have at most (1 - ε) ex(n, C4) edges. This resolves a conjecture of Balogh, Bollobás, and Simonovits for the 4-cycle.

AB - A graph is called H-free if it contains no copy o f H. Let ex(n, H) denote the Turán number for H, i.e., the maximum number of edges that an n-vertex H-free graph may have. An old result of Kleitman and Winston states that there are 2o(ex(n,C4)) C4-free graphs on n vertices. Füredi showed that almost all C4-free graphs of order n have at least cex(n, C4) edges for some positive constant c. We prove that there is a positive constant e such that almost all C 4-free graphs have at most (1 - ε) ex(n, C4) edges. This resolves a conjecture of Balogh, Bollobás, and Simonovits for the 4-cycle.

KW - Asymptotic graph enumeration

KW - Asymptotic graph structure

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KW - Extremal graphs

KW - Turán's problem

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Almost all C₄-free graphs have fewer than (1 - ε)ex(n, C₄) edges

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Keywords

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