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, $r\geq 3$, with nonnegative Lin--Lu--Yau curvature has edge-connectivity equal to its minimum incidence degree. Both the uniformity and linearity assumptions are essential. On the one hand, for every $r\geq 3$ and every integer $t\geq 2$, we construct a finite connected simple nonlinear $r$-uniform hypergraph with positive Lin--Lu--Yau curvature such that its edge-connectivity is $t$ less than its minimum degree. On the other hand, for every integer $t\geq 1$, we construct a finite connected simple linear nonuniform hypergraph with positive Lin--Lu--Yau curvature such that its edge-connectivity is also $t$ less than its minimum degree. Consequently, neither uniformity nor linearity alone is sufficient for the edge-connectivity rigidity in the hypergraph setting.