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

Grannell, M. J. and Knor, M. (2012). On the number of triangular embeddings of complete graphs and complete tripartite graphs. Journal of Graph Theory, 69(4) pp. 370–382.

DOI: https://doi.org/10.1002/jgt.20590

Abstract

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 $K_z$.

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