The genus of the n ‐octahedron: Regular cases

The n ‐octahedron O n , also denoted K (2,2, …, 2) or K n(2) , is the complete n ‐partite graph with two vertices in each partite set. The formula\documentclass{article}\pagestyle{empty}\begin{document}$$ (O_n) = \{ \frac{{(n - 1)(n - 3)}}{3}\} $$\end{document} for the (orientable) genus of O n is conjectured for all n and proved for n ≠ (mof 3). Triangular embeddigns are possible precisely when n ≠ 2 (mod 3), and the formula is established by exhibiting such embeddings.