Sidon sets and graphs without 4-cycles

The problem of determining the maximum number of edges in an $n$-vertex graph that does not contain a 4-cycle has a rich history in extremal graph theory. Using Sidon sets constructed by Bose and Chowla, for each odd prime power q we construct a graph with $q^2 - q - 2$ vertices that does not contain a 4-cycle and has at least $\frac{1}{2} q^3 - q^2 - O(q^{3/4})$ edges. This disproves a conjecture of Abreu, Balbuena, and Labbate concerning the Turán number $\mathrm{ex}(q^2 - q - 2, C_4)$.

Keywords

Turan number, Sidon set, 4-cycle

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Published 20 August 2014