### Complete regular dessins and skew-morphisms of cyclic groups

#### Abstract

A dessin is a 2-cell embedding of a connected 2-coloured bipartite graph into an orientable closed surface. A dessin is regular if its group of orientation- and colour-preserving automorphisms acts regularly on the edges. In this paper we study regular dessins whose underlying graph is a complete bipartite graph *K*_{m, n}, called (*m*, *n*)-complete regular dessins. The purpose is to establish a rather surprising correspondence between (*m*, *n*)-complete regular dessins and pairs of skew-morphisms of cyclic groups. A skew-morphism of a finite group *A* is a bijection *φ*: *A* → *A* that satisfies the identity *φ*(*x**y*) = *φ*(*x*)*φ*^{π(x)}(*y*) for some function *π*: *A* → ℤ and fixes the neutral element of *A*. We show that every (*m*, *n*)-complete regular dessin **D** determines a pair of reciprocal skew-morphisms of the cyclic groups ℤ_{n} and ℤ_{m}. Conversely, **D** can be reconstructed from such a reciprocal pair. As a consequence, we prove that complete regular dessins, exact bicyclic groups with a distinguished pair of generators, and pairs of reciprocal skew-morphisms of cyclic groups are all in a one-to-one correspondence. Finally, we apply the main result to determining all pairs of integers *m* and *n* for which there exists, up to interchange of colours, exactly one isomorphism class of (*m*, *n*)-complete regular dessins. We show that the latter occurs precisely when every group expressible as a product of cyclic groups of order *m* and *n* is abelian, which eventually comes down to the condition gcd (*m*, *ϕ*(*n*)) = gcd (*ϕ*(*m*), *n*) = 1, where *ϕ* is Euler’s totient function.

#### Keywords

DOI: https://doi.org/10.26493/1855-3974.1748.ebd

ISSN: 1855-3974

Issues from Vol 6, No 1 onward are partially supported by the Slovenian Research Agency from the Call for co-financing of scientific periodical publications