Edge connectivity of simplicial polytopes

TL;DR

This paper establishes a lower bound on the size of nontrivial minimum edge cuts in the graphs of simplicial polytopes of dimension three and higher, revealing their edge connectivity properties.

Contribution

It proves a sharp lower bound on minimum edge cuts in simplicial polytope graphs and constructs examples showing the bound is optimal.

Findings

Graphs of simplicial polytopes are highly edge-connected.

Minimum edge cuts in these graphs are either trivial or of size at least d(d+1)/2.

The bound is tight for dimensions d ≥ 4.

Abstract

We show that the graph of a simplicial polytope of dimension $d \ge 3$ has no nontrivial minimum edge cut with fewer than $d(d+1)/2$ edges, hence the graph is $\min\{\delta, d(d+1)/2\}$-edge-connected where $\delta$ denotes the minimum degree. When $d = 3$, this implies that every minimum edge cut in a plane triangulation is trivial. When $d \ge 4$, we construct a simplicial $d$-polytope whose graph has a nontrivial minimum edge cut of cardinality $d(d+1)/2$, proving that the aforementioned result is best possible.