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

Abstract

We construct several families of genus embeddings of near-complete graphs using index 2 current graphs. In particular, we give the first known minimum genus embeddings of certain families of octahedral graphs, solving a longstanding conjecture of Jungerman and Ringel, and Hamiltonian cycle complements, making partial progress on a problem of White. Index 2 current graphs are also applied to various cases of the Map Color Theorem, in some cases yielding significantly simpler solutions, e.g., the nonorientable genus of $K_{12s+8}-K_2$. We give a complete description of the method, originally due to Jungerman, from which all these results were obtained.Comment: 23 pages, 21 figures; fixed 2 figures from previous versio