Linear hypermaps--modelling linear hypergraphs on surfaces

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.