
Mike J Grannell

Thomas A McCourt
Keywords:
Orientable closed 2cell embeddings
Abstract
For all $m\geq 1$ and $k\geq 2$, we construct closed 2cell embeddings of the complete graph $K_{8km+4k+1}$ with faces of size $4k$ in orientable surfaces. Moreover, we show that when $k\geq 3$ there are at least $(2m1)!/2(2m+1)=2^{2m\text{log}_2m\mathrm{O}(m)}$ 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.