Enumerating Partial Duals of Hypermaps by Genus

TL;DR

This paper develops a genus formula for hypermaps' partial duals, generalizes existing map results, and enumerates their partial dual Euler-genus distributions using specific hypermap operations.

Contribution

It introduces an Euler-genus formula for hypermaps' partial duals, extending previous map results, and provides enumeration methods for their genus distributions.

Findings

Euler-genus formula for hypermaps' partial duals

Enumeration of partial dual Euler-genus distributions

Polynomial computations for specific hypermap classes

Abstract

The concept of partial duality in hypermaps was introduced by Chmutov and Vignes-Tourneret, and Smith independently. This notion serves as a generalization of the concept of partial duality found in maps. In this paper, we first present an Euler-genus formula concerning the partial duality of hypermaps, which serves as an invariant related to the result obtained by Chmutov and Vignes-Tourneret. This formulation also generalizes the result of Gross, Mansour, and Tucker regarding partial duality in maps. Subsequently, we enumerate the distribution of partial dual Euler-genus for hypermaps and compute the corresponding polynomial for specific classes of hypermaps through three operations: join, bar-amalgamation, and subdivision.