  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.
