Abstract
We show that a complete bipartite graph \({{\bf K}_{{p^e}, p_{f}}}\), where p is an odd prime, has an edge-transitive embedding in an orientable surface with all faces bounded by simple cycles if and only if e = f. There are exactly \({p^{2(e-1)}}\) such embeddings up to isomorphism. Among them, \({p^{e-1}}\) are orientably regular, one of which is reflexible and \({p^{e-1} -1}\) form chiral pairs. The remaining \({p^{2(e-1)} - p^{e-1}}\) embeddings are non-regular (not arc-transitive). All of these embeddings have genus\({\frac{1}{2} (p^{e}-1) (p^{e}-2)}\).
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This research was supported by NNSF (11501497, 11231008, 11661082) and a Yunnan Applied Basic Research Projects (2015FD013).
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Fan, W., Li, C.H. & Qu, H.P. A Classification of Orientably Edge-Transitive Circular Embeddings of \({{\rm K}_{{p^e}, p^{f}}}\). Ann. Comb. 22, 135–146 (2018). https://doi.org/10.1007/s00026-018-0373-5
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DOI: https://doi.org/10.1007/s00026-018-0373-5