Triangle-free planar graphs with at most [formula omitted] 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 215n/log2⁡n 3-colorings. We improve his construction, giving examples of such graphs with at most 64nlog9/2⁡3<64n0.731 3-colorings. We conjecture this exponent is optimal.