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 $k\geq 3$ 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.
Highlights
Introduction and motivationConsider an embedding of a simple graph G in an orientable surface
If each of the faces of the embedding is homeomorphic to an open disk, the embedding is said to be a 2-cell embedding of G
If the faces of a 2-cell embedding of a simple graph G with no degree one vertices are all of size less than six, the embedding is necessarily closed
Summary
Consider an embedding of a simple graph G in an orientable surface. If each of the faces of the embedding is homeomorphic to an open disk, the embedding is said to be a 2-cell embedding of G. In [12], Korzhik and Voss constructed 24m−1 nonisomorphic 2-cell orientable embeddings, where all the faces have size four, of the complete graph K8m+5, i.e. quadrangular embeddings. By applying Theorem 3.1 of [6] to the results of [5] and [7], a lower bound of the form nan on the number of nonisomorphic n-2CS(n) embeddings may be obtained for certain values of n For both n-2CS(n) and n-2CS(2n + 1) embeddings with n 6, it is non-trivial to ensure that the embeddings are closed 2-cell embeddings.
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