Identify extremal configurations on higher-genus surfaces

Identify the bounded local configuration around the dominant vertices that determines extremal face hypergraphs of \(k\)-angulations embedded on a surface of Euler genus \(\gamma\ge1\), for every \(k\ge3\).

Background

The paper explains that the planar arguments rely on non-embeddability of $K_{3,3}$. On surfaces of positive Euler genus, analogous structural bounds depend on $\gamma$, and the authors expect the dominant vertices to persist while only a bounded local structure changes around them. The precise extremal configuration has not been determined.