Triangle-free induced subgraphs of polarity graphs

TL;DR

This paper studies the maximum size of triangle-free induced subgraphs within polarity graphs derived from finite projective planes, generalizing previous bounds and providing computational insights related to an open conjecture.

Contribution

The authors extend known bounds on triangle-free induced subgraphs from specific polarity graphs to all polarity graphs, and include computational results relevant to an unresolved conjecture.

Findings

Maximum size of triangle-free induced subgraphs is generalized to all polarity graphs.

Established bounds are consistent with previous results for specific cases.

Computational experiments provide evidence related to an open conjecture.

Abstract

Given a finite projective plane $\Pi$ and a polarity $\theta$ of $\Pi$, the corresponding polarity graph is the graph whose vertices are the points of $\Pi$. Two distinct vertices $p$ and $p'$ are adjacent if $p$ is incident to $\theta (p')$. Polarity graphs have been used in a variety of extremal problems, perhaps the most well-known being the Tur\'{a}n number of the cycle of length four. We investigate the problem of finding the maximum number of vertices in an induced triangle-free subgraph of a polarity graph. Mubayi and Williford showed that when $\Pi$ is the projective geometry $PG(2,q)$ and $\theta$ is the orthogonal polarity, an induced triangle-free subgraph has at most $\frac{1}{2}q^2 + O(q^{3/2})$ vertices. We generalize this result to all polarity graphs, and provide some interesting computational results that are relevant to an unresolved conjecture of Mubayi and Williford.