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arXiv · 1702.00588

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

Zdeněk Dvořák·Jean-Sébastien Sereni
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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 $α$ 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 $α|A|$ 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.

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Zdeněk Dvořák, Jean-Sébastien Sereni. 2017-09-19. Do triangle-free planar graphs have exponentially many 3-colorings?. https://arxiv.org/abs/1702.00588

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