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.

中文翻译：

---

最多具有 64n0.731 个 3 色的无三角形平面图

Thomassen 猜想无三角形平面图的 3 着色指数呈指数增长。最近，他通过提供具有n个顶点且至多为$2^{15n/{\mathrm{日志}}_{2}n}$3 着色。我们改进了他的结构，最多给出这样的图的例子${64}^{n^{{\mathrm{日志}}_{9/2}3}}<{64}^{n^{0.731}}$3 着色。我们推测这个指数是最优的。