The genus of the n-octahedron: Regular cases

The n-octahedron On, also denoted K(2,2, …, 2) or Kn(2), is the complete n-partite graph with two vertices in each partite set. The formula for the (orientable) genus of On 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.