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We construct several families of minimum genus embeddings of dense graphsusing index 2 current graphs. In particular, we complete the genus formula forthe octahedral graphs, solving a longstanding conjecture of Jungerman andRingel, and find triangular embeddings of complete graphs minus a Hamiltoniancycle, making partial progress on a problem of White. Index 2 current graphsare also applied to various cases of the genus of the complete graphs, in somecases yielding simpler solutions, e.g., the nonorientable genus of$K_{12s+8}-K_2$. In addition, we give a topological proof of a theorem ofJungerman that shows that a symmetric type of such current graphs might notexist roughly "half of the time."

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