The Edge-Connectivity of Vertex-Transitive Hypergraphs

TL;DR

This paper extends Mader's theorem from graphs to hypergraphs, proving that connected linear uniform vertex-transitive hypergraphs are maximally edge-connected, and explores the limits of this property when conditions are relaxed.

Contribution

It generalizes a classic connectivity theorem from graphs to hypergraphs and identifies the conditions necessary for the property to hold.

Findings

Connected linear uniform vertex-transitive hypergraphs are maximally edge-connected.

Relaxing linear or uniform conditions can produce vertex-transitive hypergraphs that are not maximally edge-connected.

The result broadens understanding of symmetry and connectivity in hypergraph structures.

Abstract

A graph or hypergraph is said to be vertex-transitive if its automorphism group acts transitively upon its vertices. A classic theorem of Mader asserts that every connected vertex-transitive graph is maximally edge-connected. We generalise this result to hypergraphs and show that every connected linear uniform vertex-transitive hypergraph is maximally edge-connected. We also show that if we relax either the linear or uniform conditions in this generalisation, then we can construct examples of vertex-transitive hypergraphs which are not maximally edge-connected.