Extremal Graphs Without 4-Cycles

TL;DR

This paper establishes an upper limit on the number of edges in C4-free graphs with a specific number of vertices, and characterizes when this bound is achieved using orthogonal polarity graphs.

Contribution

It provides a new upper bound for edges in C4-free graphs on q^2 + q vertices and links the bound to orthogonal polarity graphs in even order planes.

Findings

Upper bound for edges in C4-free graphs on q^2 + q vertices.

Achievability of the bound via orthogonal polarity graphs.

Characterization for even q cases.

Abstract

We prove an upper bound for the number of edges a C4-free graph on q^2 + q vertices can contain for q even. This upper bound is achieved whenever there is an orthogonal polarity graph of a plane of even order q.