Abstract
We describe non-orientable, octagonal embeddings for certain 4-valent, bipartite Cayley graphs of finite metacyclic groups, and give a class of examples for which this embedding realizes the non-orientable genus of the group. This yields a construction of Cayley graphs for which\(2\gamma - \tilde \gamma \) is arbitrarily large, where γ and\(\tilde \gamma \) are the orientable genus and the non-orientable genus of the Cayley graph.
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