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A longstanding open question of Archdeacon and Craft asks whether everycomplete graph has a minimum genus embedding with at most one nontriangularface. We exhibit such an embedding for each complete graph except $K_8$, thecomplete graph on 8 vertices, and we go on to prove that no such embedding canexist for this graph. Our approach also solves a more general problem, giving acomplete characterization of the possible face distributions (i.e. the numbersof faces of each length) realizable by minimum genus embeddings of eachcomplete graph. We also tackle analogous questions for nonorientable andmaximum genus embeddings.

Face distributions of embeddings of complete graphs