On the number of triangular embeddings of complete graphs and complete tripartite graphs

We prove that for every prime number $p$ and odd $m>1$, as $s\to\infty$, there are at least $w^{w^2\big(\frac 1{p^4m^2}-o(1)\big)}$ face 2-colourable triangular embeddings of $K_{w,w,w}$, where $w=m\cdot p^s$. For both orientable and nonorientable embeddings, this result implies that for infinitely many infinite families of $z$, there is a constant $c>0$ for which there are at least $z^{cz^2}$ nonisomorphic face 2-colourable triangular embeddings of $K_z$.