Convex and discrete geometry Polytopes and polyhedra Theorem

Balinski's theorem

Builds on Convex polytope k-vertex-connected graph Graph

Statement

Let $d \ge 1$ and let $P$ be a convex polytope of dimension $d$ . Let $G(P)$ be the graph whose vertices are the vertices of $P$ and whose edges are the edges of $P$ . Then $G(P)$ is d-vertex-connected: it has more than $d$ vertices, and it stays connected after the removal of any $d - 1$ vertices.

Remarks

The theorem was proved by Michel Balinski in 1961. Proof idea: remove a set $S$ of $d - 1$ vertices, and pass a hyperplane through $S$ and one further vertex. From every vertex that does not maximise a linear function on $P$ there is an edge along which the function increases, as in the simplex method. Use this for the linear function defining the hyperplane and for its negative. It connects every remaining vertex, avoiding $S$ , to the top face or the bottom face of $P$ . These pieces are then joined through the extra vertex.

Example: the graph of the $3$ -cube is $3$ -connected; removing the $3$ neighbours of a vertex isolates it, so the bound $d$ is sharp.

Going deeper

For a student

For $d = 3$ this is the easy half of Steinitz's theorem on graphs of $3$ -polytopes. The bound is sharp for every simple polytope, since a vertex has only $d$ neighbours. It is one of the few general facts known about graphs of polytopes in high dimension, along with results and open questions about their diameter, such as the Hirsch conjecture.

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Tag	`YBNZWCG8AC`
URI	`m2://YBNZWCG8AC`
Kind	statement, theorem
Writing	W0 stub
MSC	52B05, 05C40
Status	This copy of the site has no build data. Run `m2 status YBNZWCG8AC`.
About	Convex polytope, k-vertex-connected graph
Uses	Convex polytope, k-vertex-connected graph, Graph
Same as	Wikidata Q32182 Wikipedia Balinski's theorem

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