2018/06/19 by Yan‐Quan Feng, Feng, Yan-Quan, Kan Hu +7 · 1 citation
Computer Science · Engineering · Mathematics · #05E18 (primary) #20B25 #57M15 (secondary) #Coding theory and cryptography #Combinatorics (math.CO) #FOS: Mathematics #Finite Group Theory Research #Group Theory (math.GR) #graph theory and CDMA systems
paper · pdf · doi:10.48550/arxiv.1806.07024
openalex publication_date 2018/06/19 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 Km,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)φπ(x)(y) for some function π\colon 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 ℤn and ℤm. 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 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.