0801.0939Jan 7, 2008Added Remark 1.2 and reference to the article [Incidence graphs of convex polytopes, J. Combin. Theory 2 (1967), 466-506] by G.T. Sallee

On the graph-connectivity of skeleta of convex polytopes

Christos A. Athanasiadis

Combinatorics

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Abstract

Given a $d$ -dimensional convex polytope $P$ and nonnegative integer $k$ not exceeding $d-1$ , let $G_k (P)$

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denote the simple graph on the node set of

k

-dimensional faces of

P

in which two such faces are adjacent if there exists a

(k+1)

-dimensional face of

P

which contains them both. The graph

G_k (P)

is isomorphic to the dual graph of the

(d-k)

-dimensional skeleton of the normal fan of

P

. For fixed values of

k

and

d

, the largest integer

m

such that

G_k (P)

is

m

-vertex-connected for all

d

-dimensional polytopes

P

is determined. This result generalizes Balinski's theorem on the one-dimensional skeleton of a

d

-dimensional convex polytope.

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Nearby in the stack

On the graph-connectivity of skeleta of convex polytopes

Claims

Methods

Limitations

Nearby in the stack

Convex hulls of spheres and convex hulls of convex polytopes lying on parallel hyperplanes

A Robust Version of Convex Integral Functionals

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