The number of colorings of planar graphs with no separating triangles

A classical result of Birkhoff and Lewis implies that every planar graph with n vertices has at least 15⋅2n−1 distinct 5-vertex-colorings. Equality holds for planar triangulations with n−4 separating triangles. We show that, if a planar graph has no separating triangle, then it has at least (2+10−12)n distinct 5-vertex-colorings. A similar result holds for k-colorings for each fixed k≥5. Infinitely many planar graphs without separating triangles have less than 2.252n distinct 5-vertex-colorings. As an auxiliary result we provide a complete description of the infinite 6-regular planar triangulations.