  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 topological proof of a theorem of
Jungerman that shows that a symmetric type of such current graphs might not
exist roughly "half of the time."
