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arXiv · 1911.05214

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

Timothy Sun
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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 $K_{12s+8}-K_2$. 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."

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Timothy Sun. 2023-06-23. Jungerman ladders and index 2 constructions for genus embeddings of dense regular graphs. https://arxiv.org/abs/1911.05214

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