Edge-connectivity and LLY curvature of hypergraphs

Abstract

Chen, Liu, and You \cite{ChenLiuYou2025} proved that a locally finite connected graph with positive Lin--Lu--Yau curvature has edge-connectivity equal to its minimum degree. Liu and Xia \cite{LiuXia2026} subsequently showed that the same conclusion holds for every finite connected graph with nonnegative Lin--Lu--Yau curvature and classified all infinite exceptions. We investigate the corresponding problem for the random-walk curvature of hypergraphs introduced by Tian and Zhao \cite{TianZhao2025}. We formulate a hypergraph analogue of the combinatorial inequality used by Liu and Xia \cite{LiuXia2026} and use it to study edge cuts in uniform linear hypergraphs. Our first main result asserts that every locally finite connected $r$-uniform linear hypergraph, $r\ge 3$, with nonnegative Lin--Lu--Yau curvature has edge-connectivity equal to its minimum incidence degree. The linearity assumption is essential. In particular, for every $r\ge 3$ and every integer $t\ge 2$, we construct a finite connected simple nonlinear $r$-uniform ypergraph with positive Lin--Lu--Yau curvature such that its edge-connectivity is $t$ less than its minimum degree. Consequently, in the nonlinear setting the gap between minimum degree and edge-connectivity can be arbitrarily large even under strictly positive curvature.