# Do triangle-free planar graphs have exponentially many 3-colorings?

## TL;DR

This paper explores Thomassen's conjecture that triangle-free planar graphs have exponentially many 3-colorings, establishing an equivalent statement involving edge deletions and 3-colorability, and proves it in certain restricted cases.

## Contribution

The paper shows the equivalence of Thomassen's conjecture to a new statement involving edge deletions and 3-colorability, and proves this in specific restricted scenarios.

## Key findings

- Thomassen's conjecture is equivalent to a statement about edge deletions and 3-colorability.
- The equivalence enables studying the conjecture in restricted cases.
- The paper proves the statement in some restricted situations.

## Abstract

Thomassen conjectured that triangle-free planar graphs have an exponential number of $3$-colorings. We show this conjecture to be equivalent to the following statement: there exists a positive real $\alpha$ such that whenever $G$ is a planar graph and $A$ is a subset of its edges whose deletion makes $G$ triangle-free, there exists a subset $A'$ of $A$ of size at least $\alpha|A|$ such that $G-(A\setminus A')$ is $3$-colorable. This equivalence allows us to study restricted situations, where we can prove the statement to be true.