Biembeddings of metacyclic groups and triangulations of orientable surfaces by complete graphs

Grannell, Mike and Knor, Martin (2012). Biembeddings of metacyclic groups and triangulations of orientable surfaces by complete graphs. Electronic Journal of Combinatorics, 19(3) P29.

URL: http://www.combinatorics.org/ojs/index.php/eljc/ar...

Abstract

For each integer $n\ge 3$, $n\ne 4$, for each odd integer $m\ge 3$, and for any $\lambda\in \mathbb{Z}_n$ of (multiplicative) order $m'$ where $m'\mid m$, we construct a biembedding of Latin squares in which one of the squares is the Cayley table of the metacyclic group $\mathbb{Z}_m\ltimes_{\lambda}\mathbb{Z}_n$. This extends the spectrum of Latin squares known to be biembeddable.

The best existing lower bounds for the number of triangular embeddings of a complete graph $K_z$ in an orientable surface are of the form $z^{z^2(a-o(1))}$ for suitable positive constants $a$ and for restricted infinite classes of $z$. Using embeddings of $\mathbb{Z}_3\ltimes_{\lambda}\mathbb{Z}_n$, we extend this lower bound to a substantially larger class of values of $z$.

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