A better bound on the largest induced forests in triangle-free planar graphs

TL;DR

This paper improves the lower bound on the size of the largest induced forest in triangle-free planar graphs from approximately 5/9.17 to 5/9 of the vertices, advancing understanding of graph structure.

Contribution

The paper presents a tighter lower bound of 5/9 on the largest induced forest size in triangle-free planar graphs, improving previous bounds through novel techniques.

Findings

Largest induced forest size is at least 5/9 of vertices.

Improved bound from 5/9.17 to 5/9.

Technique inspired by recent ideas from Lukot'ka, Mazák, and Zhu.

Abstract

It is well-known that there exists a triangle-free planar graph of $n$ verticess such that the largest induced forest has size at most $\frac{5n}{8}$. Salavatipour proved that there is a forest of size at least $\frac{5n}{9.41}$ in any triangle-free planar graph of $n$ vertices. Dross, Montassier and Pinlou improved Salavatipour's bound to $\frac{5n}{9.17}$. In this work, we further improve the bound to $\frac{5n}{9}$. Our technique is inspired by the recent ideas from Lukot'ka, Maz{\'a}k and Zhu.