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Complete Circular Regular Dessins of Type \(\{2^e,2^f\}\) I: Metacyclic Case

Abstract

A complete dessin of type \(\{2^e,2^f\}\) is an orientable map with underlying graph being a complete bipartite graph \(\mathbf{K}_{2^e,2^f}\), which is said to be regular if all edges are equivalent under the group of color- and orientation-preserving automorphisms, and circular if the boundary cycle of each face is a circuit (a simple cycle). As one of a series papers towards a classification of complete circular regular dessins of type \(\{m,n\}\), this paper presents such a classification for the case \(\{m,n\}=\{2^e,2^f\}\), where ef are positive integers and \(e\ge f\ge 2\). We note that the group G of color- and orientation-preserving automorphisms is a bicyclic 2-group of type \(\{2^e,2^f\}\), and our analysis splits naturally into two cases depending on whether the group G is metacyclic or not. In this paper, we deal with the case that G is metacyclic.

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Correspondence to Wenwen Fan.

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Communicated by Kolja Knauer.

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This work was partially supported by NSFC: 12061092, 11501497; Yunnan Applied Basic Research Projects: 202101AT070137.

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Fan, W. Complete Circular Regular Dessins of Type \(\{2^e,2^f\}\) I: Metacyclic Case. Ann. Comb. 26, 125–144 (2022). https://doi.org/10.1007/s00026-021-00562-3

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Mathematics Subject Classification

  • 20B15
  • 20B30
  • 05C25