Duality on hypermaps with symmetric or alternating monodromy group

Abstract

Duality is the operation that interchanges hypervertices and hyperfaces on oriented hypermaps. The duality index measures how far a hypermap is from being self-dual. We say that an oriented regular hypermap has \emph{duality-type} $\{l,n\}$ if $l$ is the valency of its vertices and $n$ is the valency of its faces. Here, we study some properties of this duality index in oriented regular hypermaps and we prove that for each pair $n$, $l \in \mathbb{N}$, with $n,l \geq 2$, it is possible to find an oriented regular hypermap with extreme duality index and of duality-type $\{l,n \}$, even if we are restricted to hypermaps with alternating or symmetric monodromy group.