A longstanding open question of Archdeacon and Craft asks whether every complete graph has a minimum genus embedding with at most one nontriangular face. We exhibit such an embedding for each complete graph except for $K_8$, and we go on to prove that no such embedding can exist for this graph. Our approach also solves a more general problem, giving a complete characterization of the possible face distributions (i.e., the numbers of faces of each length) realizable by minimum genus embeddings of each complete graph. We also tackle analogous questions for nonorientable and maximum genus embeddings.

中文翻译：

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完整图的嵌入的面分布

关于Archdeacon和Craft的一个长期未解决的问题是，每个完整图形是否都具有最少的类嵌入到最多一个非三角形的面。除了每个完整的图，我们都展示了这样的嵌入${ķ}_8$，然后我们继续证明该图不存在这种嵌入。我们的方法还解决了一个更普遍的问题，即通过每个完整图形的最小属类嵌入实现可实现的人脸分布（即每个长度的人脸数量）的完整表征。我们还针对不可定向的最大属类嵌入处理类似的问题。