A new proof of Balinski's theorem on the connectivity of polytopes

TL;DR

This paper offers a new proof of Balinski's theorem, demonstrating that the graph of a d-dimensional convex polytope is d-connected, with detailed insights into separating sets of size d.

Contribution

The paper presents a novel proof of Balinski's theorem, including new insights into the structure of separating sets of size d.

Findings

Proof confirms d-connectedness of polytope graphs

Provides detailed analysis of separating sets of size d

Some aspects of separating sets are newly identified

Abstract

Balinski (1961) proved that the graph of a $d$-dimensional convex polytope is $d$-connected. We provide a new proof of this result. Our proof provides details on the nature of a separating set with exactly $d$ vertices; some of which appear to be new.