Doubly even orientable closed 2-cell embeddings of the complete graph.

Grannell, Mike and McCourt, Thomas (2014). Doubly even orientable closed 2-cell embeddings of the complete graph. Electronic Journal of Combinatorics, 21(1), article no. P1.22.

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

Abstract

For all $m\geq 1$ and $k\geq 2$ , we construct closed 2-cell embeddings of the complete graph $K_{8km+4k+1}$ with faces of size $4k$ in orientable surfaces. Moreover, we show that when $k\geq3$ there are at least $(2m-1)!/2(2m+1)=2^{2m\text{log}_2m-\mathrm{O}(m)}$ nonisomorphic embeddings of this type. We also show that when $k=2$ there are at least $\frac14 \pi^{\frac12}m^{-\frac{5}{4}}\left(\frac{4m}{e^2}\right)^{\sqrt{m}}{(1-\mathrm{o}(1))}$ nonisomorphic embeddings of this type.