# A lower bound on the order of the largest induced linear forest in triangle-free planar graphs

## TL;DR

This paper establishes a lower bound on the size of the largest induced linear forest in triangle-free planar graphs, providing both a theoretical guarantee and examples showing the bound's tightness.

## Contribution

It introduces a new lower bound for the largest induced linear forest in triangle-free planar graphs and demonstrates the bound's near-optimality with explicit constructions.

## Key findings

- Every triangle-free planar graph has an induced linear forest of at least (9n - 2m)/11 vertices.
- The lower bound simplifies to at least (5n + 8)/11 vertices.
- Existence of graphs with largest induced linear forest close to n/2.

## Abstract

We prove that every triangle-free planar graph of order $n$ and size $m$ has an induced linear forest with at least $\frac{9n - 2m}{11}$ vertices, and thus at least $\frac{5n + 8}{11}$ vertices. Furthermore, we show that there are triangle-free planar graphs on $n$ vertices whose largest induced linear forest has order $\lceil \frac{n}{2} \rceil + 1$.