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.
Highlights
A classical theorem of Grotzsch [5] asserts that every triangle-free planar graph is 3-colorable
The new proofs are simpler than the original argument, and often target further developments — algorithmic aspects or extension to other surfaces. Refining some of his arguments, Thomassen [12] established that every planar graph of girth at least five has exponentially many — in terms of the number of vertices — list colorings provided all lists have size at least three
It could still be true that triangle-free planar graphs admit exponentially many 3-colorings
Summary
A classical theorem of Grotzsch [5] asserts that every triangle-free planar graph is 3-colorable. (TRIA) There is a positive real number α such that for every planar graph G and every subset X of edges such that G − X is triangle-free, there exists a 3coloring c of G−X such that at least α|X| edges in X join vertices of different colors under c It suffices to subdivide each edge in X by a vertex placed in R= to see that (TRIA) is implied by (RGEN). Since the vertices in S ∪ T are incident with the same face of G , this is reminiscent of a well-known result of Thomassen [9] (Theorem 15 below), which implies that such a coloring exists whenever G has girth at least 5 and S ∪ T is an independent set As it turns out, the graph G can have 4-cycles, but these are relatively easy to deal with (we can eliminate separating 4-cycles via a precoloring extension argument, and 4-faces can be reduced in a standard way by collapsing).
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