Complete regular dessins and skew-morphisms of cyclic groups - Feng, Yan-Quan (Ars mathematica contemporanea, 2020)
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 ?$φ\colon A→ A$? that satisfies the identity ?$φ(xy)=φ(x)φ^{π(x)}(y)$? for some function ?$π\colon A→\mathbb{Z}$? and fixes the neutral element of ?$A$?. We show that every ?$(m,n)$?-complete regular dessin ?$\mathcal{D}$? determines a pair of reciprocal skew-morphisms of the cyclic groups ?$\mathbb{Z}_n$? and ?$\mathbb{Z}_m$?. Conversely, ?$\mathcal{D}$? 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 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 ?$(m,n)$?-complete regular dessin. 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: biciklična grupa; bicyclic group; graph embedding; poševni morfizem; pravilna risba; regular dessin; skew-morphism; vložitev grafa