Sidon sets and $C_4$-saturated graphs

TL;DR

This paper explores the use of Sidon sets in constructing $C_4$-free and $C_4$-saturated graphs, verifying a conjecture and providing new graph constructions with specific properties.

Contribution

It verifies a conjecture about the number of $C_4$ copies in near-extremal graphs and introduces new $C_4$-saturated graph constructions using Sidon sets.

Findings

Verified Erd"os and Simonovits conjecture for Sidon set-based graphs.

Provided sufficient conditions for $C_4$-saturation in sum graphs.

Described new classes of $C_4$-saturated graphs.

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.