Enumerating Partial Duals of Hypermaps by Genus
Wenwen Liu, Yichao Chen
TL;DR
This work extends partial duality from maps to hypermaps by embedding hypermaps on surfaces via bi-rotation systems, arrow presentations, and bipartite graph models, and then derives a robust Euler-genus framework. It defines the partial-dual hypermap $H^A$ for $A\subseteq E(H)$, establishes invariants such as $v(H^A)=f(A)$ and $e(H^A)=e(H)$, and introduces the partial-dual Euler-genus and orientable-genus polynomials, which encode genus changes across all partial duals. The paper develops three constructive operations—join, bar-amalgamation, and subdivision—and provides exact relations for Euler characteristic, Euler-genus, and the corresponding polynomials under these operations, enabling enumeration of partial-duals in composite hypermaps. Applications to hypertrees and hyper-ladder maps yield explicit polynomial formulas, including $\partial_{\varepsilon_T}(z)$ for hypertrees and $\partial_{\varepsilon_{H_n}}(z)=2(1+z^2)^{n-1}$ for hyper-ladder families, highlighting both interpolating and non-interpolating spectra phenomena. Overall, the results furnish tools for systematic genus analysis and polynomial enumeration of partial-duals in hypermaps, generalizing known map results to a broader hypergraph setting with potential combinatorial and topological applications.
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.