Biembeddings of metacyclic groups and triangulations of orientable surfaces by complete graphs (Q456361)

Summary: For each integer \(n \geq 3\), \(n \neq 4\), for each odd integer \(m \geq 3\), 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 \(z\).