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

2019/11/13 by Sun, Timothy · 2 citations
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.1911.05214

Abstract

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-K₂. 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."

Cited by

• Revisiting Cases 2 and 11 of the Map Color Theorem
• Orientable triangular embeddings of the complete graphs on 36s vertices from noncyclic current graphs

Related

• Orientable triangular embeddings of the complete graphs on 36s vertices from noncyclic current graphs
• Genus embeddings of complete graphs minus a matching
• Settling the nonorientable genus of the nearly complete bipartite graphs
• Total embedding distributions of Ringel ladders
• Face distributions of embeddings of complete graphs