# Flexibility of planar graphs of girth at least six

## TL;DR

This paper proves that planar graphs with girth at least six and list sizes of three or more can be colored to respect a constant fraction of vertex preferences, advancing understanding of list coloring flexibility.

## Contribution

It establishes a new lower bound on the fraction of preferences that can be respected in list coloring of planar graphs with girth at least six.

## Key findings

- Existence of an L-coloring respecting a constant fraction of preferences
- Improved bounds for list coloring in planar graphs with girth ≥ 6
- Advancement in understanding coloring flexibility under constraints

## 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 has girth at least six and all lists have size at least three, then there exists an L-coloring respecting at least a constant fraction of the preferences.