On the number of nonisomorphic orientable regular embeddings of complete graphs
In this paper those 2-cell orientable embeddings of a complete graph \(K_{n+1}\) are considered which are generated by rotation schemes on an abelian group \(\Phi\) of order \(n+1\), where a rotation scheme on \(\Phi\) is defined as a cyclic permutation \((\beta_1,\beta_2,\dots, \beta_n)\) of all nonzero elements of \(\Phi\). It is shown that two orientable embeddings of \(K_{n+1}\) generated by schemes \((\beta_1,\beta_2,\dots, \beta_n)\) and \((\gamma_1,\gamma_2,\dots, \gamma_n)\) are isomorphic if and only if \((\gamma_1, \gamma_2,\dots, \gamma_n)= (\varphi(\beta_1), \varphi(\beta_2),\dots, \varphi(\beta_n))\) or \((\gamma_1,\gamma_2,\dots, \gamma_n)= (\varphi(\beta_n),\dots, \varphi(\beta_2), \varphi(\beta_1))\), where \(\varphi\) is an automorphism of \(\Phi\). As a consequence, representing schemes by index one current graphs, the following results are obtained. For every \(s\geq 1\) the graphs \(K_{12s+4}\) and \(K_{12s+7}\) have no less than \(4^s\) mutually nonisomorphic orientable 3-face-colorable triangular embeddings. For every \(s\geq 1\) the graph \(K_{8s+5}\) has no less than \(14\cdot 8^{s-1}\) mutually nonisomorphic orientable quadrangular embeddings.
scientific article; zbMATH DE number 1843738