A 
-polytopal graph is a graph 
for which there exists a 
-dimensional convex polytope 
such that the vertices
and edges of 
are in a one-to-one incidence-preserving
correspondence with those of 
. In other words 
is 
-polytopal iff it is isomorphic to
the 1-skeleton of a convex
polytope 
of dimension 
. If 
, the graph is called a polyhedral
graph.
Pineda-Villavicencio (2026) proved that, for every 
, every integer 
, and every 
, there is a simplicial
polytope of dimension 
whose graph has vertex
connectivity exactly 
, has graph toughness less
than 
,
and has no Hamiltonian path. Thus arbitrarily
high vertex connectivity does not force a
Hamiltonian cycle in these graphs.
Pineda-Villavicencio (2026) reports using GPT-6 Astra to explore research strategies,
while independently verifying the mathematics.
