Edge connectivity of simplicial polytopes

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.