Exponentially many nonisomorphic orientable triangular embeddings of \(K_{12s+3}\)
The orientable case \(n\equiv\pmod{12}\) of the Heawood Map Color Theorem was solved by using, for each \(s\), an index three current graph with current group \(Z_{12s+3}\) (see \textit{G. Ringel} [''Map Color Theorem,'' Springer-Verlag, Berlin (1974; Zbl 0287.05102)]), to give an orientable triangular imbedding of \(K_{12s+3}\). The present author provides a new index three current graph, with the same current group and the same result. He then shows that, for \(s\geq 11\), there are at least \(2^{2s-11}\) nonisomorphic orientable triangular imbeddings of \(K_{12s+3}\), thus completing the proof that there are constants \(M\), \(c> 0\), and \(b\geq 1/12\) such that for every \(n\geq M\) there are at least \(c2^{bn}\) nonisomorphic orientable as well as nonorientable genus imbeddings of the complete graph \(K_n\).
scientific article; zbMATH DE number 5542380