Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs

We construct several families of minimum genus embeddings of dense graphs using index 2 current graphs. In particular, we complete the genus formula for the octahedral graphs, solving a longstanding conjecture of Jungerman and Ringel, and find triangular embeddings of complete graphs minus a Hamiltonian cycle, making partial progress on a problem of White. Index 2 current graphs are also applied to various cases of the genus of the complete graphs, in some cases yielding simpler solutions, e.g., the nonorientable genus of $K_{12s+8}-K_2$. In addition, we give a simpler proof of a theorem of Jungerman that shows that a symmetric type of such current graphs might not exist roughly “half of the time”.