Almost all C 4 - free graphs have less than ( 1 − ε ) ex ( n , C 4 ) edges
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چکیده
A graph is called H-free if it contains no copy of H. An old result of Kleitman and Winston [12] states that there are 2Θ(n 3/2) C4-free graphs on n vertices. Füredi [8] showed that almost all C4-free graphs of order n have at least c · ex(n, C4) edges for some positive constant c > 0. We prove that there is an ε > 0 such that almost all C4-free graphs have at most (1 − ε) · ex(n, C4) edges. This resolves a conjecture of Balogh, Bollobás and Simonovits [4] for the 4-cycle. Mathematics subject classification: 05C35, 05C30, 05D40, 05A16
منابع مشابه
Almost all C 4 - free graphs have fewer than ( 1 − ε ) ex ( n , C 4 ) edges
A graph is called H-free if it contains no copy of 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 c ex(n,C4) edges for some positive constant c. We pro...
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