# Flexibility of triangle-free planar graphs

## TL;DR

This paper proves that triangle-free planar graphs with list sizes of at least four can be colored to respect a constant fraction of vertex preferences, advancing understanding of list coloring flexibility.

## Contribution

It establishes a lower bound on the fraction of preferences that can be respected in list coloring of triangle-free planar graphs.

## Key findings

- Existence of an L-coloring respecting a constant fraction of preferences
- Applicable to triangle-free planar graphs with list sizes ≥ 4
- Advances list coloring theory for planar graphs

## Abstract

Let G be a planar graph with a list assignment L. Suppose a preferred color is given for some of the vertices. We prove that if G is triangle-free and all lists have size at least four, then there exists an L-coloring respecting at least a constant fraction of the preferences.