Triangle-free planar graphs with at most $64^{n^{0.731}}$ 3-colorings

TL;DR

This paper improves the construction of triangle-free planar graphs with a limited number of 3-colorings, reducing the upper bound from previous work and proposing the conjecture that this bound is optimal.

Contribution

It provides a tighter upper bound on the number of 3-colorings in such graphs, advancing understanding of their coloring properties.

Findings

Constructed graphs with at most 64^{n^{0.731}} 3-colorings

Improved previous upper bounds on 3-colorings

Conjecture that this bound is optimal

Abstract

Thomassen conjectured that triangle-free planar graphs have exponentially many 3-colorings. Recently, he disproved his conjecture by providing examples of such graphs with $n$ vertices and at most $2^{15n/\log_2 n}$ 3-colorings. We improve his construction, giving examples of such graphs with at most $64^{n^{log_{9/2} 3}}<64^{n^{0.731}}$ 3-colorings. We conjecture this exponent is optimal.