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

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.