Triangle-free induced subgraphs of the unitary polarity graph

TL;DR

This paper investigates the maximum size of triangle-free induced subgraphs in the unitary polarity graph of finite projective planes, providing bounds and constructions, especially for the Desarguesian case when q is even.

Contribution

It establishes an upper bound on the size of triangle-free induced subgraphs in the unitary polarity graph and constructs near-optimal examples for certain cases.

Findings

Maximum size of triangle-free induced subgraphs is at most (q^4+q)/2.

For Desarguesian planes with even q, the bound is nearly achieved with constructions.

Discusses special cases like the Figueroa plane.

Abstract

Let $\perp$ be a unitary polarity of a finite projective plane $\pi$ of order $q^2$. The unitary polarity graph is the graph with vertex set the points of $\pi$ where two vertices $x$ and $y$ are adjacent if $x \in y^\perp$. We show that a triangle-free induced subgraph of the unitary polarity graph of an arbitrary projective plane has at most $(q^4+q)/2$ vertices. When $\pi$ is the Desarguesian projective plane $\mathrm{PG}(2,q^2)$ and $q$ is even, we show that the upper bound is asymptotically sharp, by providing an example on $q^4/2$ vertices. Finally, the case when $\pi$ is the Figueroa plane is discussed.