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arXiv · 1810.05262

Sidon sets and $C_4$-saturated graphs

David Fernando Daza·Carlos Alberto Trujillo·Fenando Andrés Benavides
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Abstract

The problem of determining the Tur\'an number of $C_4$ is a well studied problem that dates back to a paper of Erd\"os from 1938. It is known that Sidon sets can be used to construct $C_4$-free graphs. If $\A$ is a Sidon set in the abelian group $X$, the sum graph $G_{X, \A}$ with vertex set $X$ and edges set $E=\{\{x, y\}:x\neq y, x+y\in \A\}$ is $C_4$-free. Using the sum graph of a Sidon set of type Singer we verify a conjecture of Erd\"os and Simonovits concerning the number of copies of $C_4$ in a graph with $ex(q^2+q+1, C_4)+1$ edges. Further, we give a sufficient condition for the sum graph of a Sidon set to be $C_4$-saturated and describe new $C_4$-saturated graphs.

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David Fernando Daza, Carlos Alberto Trujillo, Fenando Andrés Benavides. 2018-10-11. Sidon sets and $C_4$-saturated graphs. https://arxiv.org/abs/1810.05262

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