Edge connectivity of simplicial polytopes

TL;DR

This paper investigates the edge connectivity of graphs of simplicial polytopes, establishing conditions under which minimum edge cuts are trivial and constructing examples with nontrivial cuts, thus advancing understanding of their combinatorial properties.

Contribution

It proves that minimum edge cuts of size up to 4d-7 are trivial in simplicial d-polytopes for d≥3 and constructs examples with nontrivial cuts, showing the bounds are tight.

Findings

Minimum edge cuts of size ≤ 4d-7 are trivial in simplicial d-polytopes for d≥3.

Graphs of simplicial d-polytopes are min{δ, 4d-6}-edge-connected.

Constructed examples of simplicial d-polytopes with nontrivial minimum edge cuts of size (d²+d)/2.

Abstract

A simplicial polytope is a polytope with all its facets being combinatorially equivalent to simplices. We deal with the edge connectivity of the graphs of simplicial polytopes. We first establish that, for any $d\ge 3$, for any $d\ge 3$, every minimum edge cut of cardinality at most $4d-7$ in such a graph is \textit{trivial}, namely it consists of all the edges incident with some vertex. A consequence of this is that, for $d\ge 3$, the graph of a simplicial $d$-polytope with minimum degree $\delta$ is $\min\{\delta,4d-6\}$-edge-connected. In the particular case of $d=3$, we have that every minimum edge cut in a plane triangulation is trivial; this may be of interest to researchers in graph theory. Second, for every $d\ge 4$ we construct a simplicial $d$-polytope whose graph has a nontrivial minimum edge cut of cardinality $(d^{2}+d)/2$. This gives a simplicial 4-polytope with a nontrivial minimum edge cut that has ten edges. Thus, the aforementioned result is best possible for simplicial $4$-polytopes.