Linear hypermaps--modelling linear hypergraphs on surfaces

TL;DR

The paper develops a theory of linear hypermaps as $2$-cell embeddings of associated graphs of linear hypergraphs on compact surfaces, encoded combinatorially by flags with three fixed-point-free involutions $r_0,r_1,r_2$ whose monodromy group $G$ acts transitively. It establishes algebraic and morphism frameworks, including Euler-type formulas $v+e+f-|\\Phi|/2=2-2g$ (orientable) or $2-g$ (non-orientable), and identifies regular linear hypermaps via $\mathrm{Aut}(\mathcal{M})\cong G$. The authors classify regular linear hypermaps on the sphere, and, using computational tools, enumerate proper orientable regular linear hypermaps of genus up to $101$ with explicit $\mathcal{M}$-sequences; they also analyze hypermaps with automorphism group $A_5\times\mathbb{Z}_2$, obtaining 19 non-isomorphic examples and distinguishing orientability. The work links hypergraph configurations to geometric embeddings, provides a robust algebraic toolkit (flags, monodromy, morphisms), and lays out open problems on broader surface classes and non-abelian simple automorphism groups, contributing to symmetry-rich combinatorial geometry on surfaces.

Abstract

A hypergraph is linear if each pair of distinct vertices appears in at most one common edge. We say $\varGamma=(V,E)$ is an associated graph of a linear hypergraph $\mathcal{H}=(V, X)$ if for any $x\in X$, the induced subgraph $\varGamma[x]$ is a cycle, and for any $e\in E$, there exists a unique edge $y\in X$ such that $e\subseteq y$. A linear hypermap $\mathcal{M}$ is a $2$-cell embedding of a connected linear hypergraph $\mathcal{H}$'s associated graph $\varGamma$ on a compact connected surface, such that for any edge $x\in E(\mathcal{H})$, $\varGamma[x]$ is the boundary of a $2$-cell and for any $e\in E(\varGamma)$, $e$ is incident with two distinct $2$-cells. In this paper, we introduce linear hypermaps to model linear hypergraphs on surfaces and regular linear hypermaps modelling configurations on the surfaces. As an application, we classify regular linear hypermaps on the sphere and determine the total number of proper regular linear hypermaps of genus 2 to 101.

Figures (8)

• Figure 1: The modification process.
• Figure 2: The Fano plane on Torus.
• Figure 3: 7 coins on the Sphere.
• Figure 4: A linear hypermap on Torus.
• Figure 5: A flag of a linear hypermap.

Theorems & Definitions (48)

• Definition 1.1
• Definition 1.2
• Remark 2.1
• Remark 2.2
• Proposition 2.3
• Proposition 2.4
• proof
• Proposition 2.5
• proof
• Definition 2.6