Triangle-free planar graphs with the smallest independence number

TL;DR

This paper improves the lower bound on the independence number for most n-vertex planar triangle-free graphs, showing they have larger independent sets than previously established, except for a specific infinite class.

Contribution

The authors establish a tighter lower bound of (n+2)/3 for the independence number in all but one infinite class of planar triangle-free graphs.

Findings

Most planar triangle-free graphs have independence number at least (n+2)/3.

A unique infinite class of graphs achieves the previous bound of (n+1)/3.

The result refines understanding of independence numbers in planar graphs.

Abstract

Steinberg and Tovey proved that every n-vertex planar triangle-free graph has an independent set of size at least (n+1)/3, and described an infinite class of tight examples. We show that all n-vertex planar triangle-free graphs except for this one infinite class have independent sets of size at least (n+2)/3.