Enumeration of plane triangulations with all vertices of degree $3$ or $6$ and a new characterization of akempic triangulations

Plane triangulations with all vertices of degree $3$ or $6$ are enumerated. A plane triangulation is said to be akempic if it has a $4$-colouring such that no two adjacent triangles have the same three colours and this colouring is not Kempe equivalent to any other colouring. Mohar (1985 and 1987) characterized and enumerated akempic triangulations with all vertices of degree $3$ or $6$. We give a new characterization of the akempic triangulations and a new proof of the Mohar enumeration theorem.