David W. Barnette

Generating the triangulations of the projective plane

Abstract Using the operations of face splitting and its dual, vertex splitting, one can generate all of the triangulations of the projective plane from two minimal triangulations. One of the minimal triangulations is the familiar embedding of the complete graph on 6 vertices. The other is a triangulation with 7 vertices.

The minimum number of vertices of a simple polytope

Ad-polytope is ad-dimensional set that is the convex hull of a finite number of points. Ad-polytope is simple provided each vertex meets exactlyd edges. It has been conjectured that for simple polytopes {fx121-1} wherefi is the number ofi-dimensional faces of the polytope. In this paper we show that inequality (i) holds for all simple polytopes.

Graph theorems for manifolds

Two basic theorems about the graphs of convex polytopes are that the graph of ad-polytope isd-connected and that it contains a refinement of the complete graph ond+1 vertices. We obtain generalizations of these theorems, and others, for manifolds. We also supply some details for a proof of the lower bound inequality for manifolds.

The triangulations of the 3-sphere with up to 8 vertices

Abstract The different combinatorial types of triangulations of the 3-sphere with up to 8 vertices are determined. Using similar methods we show that one cannot always preassign the shape of a facet of a 4-polytope.

All 2-manifolds have finitely many minimal triangulations

A triangulation of a 2-manifoldM is said to be minimal provided one cannot produce a triangulation ofM with fewer vertices by shrinking an edge. In this paper we prove that all 2-manifolds have finitely many minimal triangulations. It follows that all triangulations of a given 2-manifold can be generated from the minimal triangulations by a process called vertex splitting.

An upper bound for the diameter of a polytope

The distance between two vertices of a polytope is the minimum number of edges in a path joining them. The diameter of a polytope is the greatest distance between two vertices of the polytope. We show that if P is a d-dimensional polytope with n facets, then the diameter of P is at most 132^d^-^3(n-d+52).

Hamiltonian circuits on 3-polytopes

Abstract The smallest number of vertices, edges, or faces of any 3-polytope with no Hamiltonian circuit is determined. Similar results are found for simplicial polytopes with no Hamiltonian circuit.

2-connected spanning subgraphs of planar 3-connected graphs

Abstract We prove that every planar 3-connected graph has a 2-connected spanning subgraph of maximum valence 15. We give an example of a planar 3-connected graph with no spanning 2-connected subgraph of maximum valence five.

Decompositions of homology manifolds and their graphs

The graph of everyd-dimensional convex polytope isd-connected and contains a refinement of the complete graph ond+1 vertices. These two theorems are generalized to pseudomanifolds and to some very general decompositions of homology manifolds.

Wv paths on 3-polytopes

Abstract A path in a polytope is called a Wv path provided it never returns to any facet once it leaves it (Theorem 1). If x and y are two vertices of a 3-dimensional convex polytope then x and y can be joined by a Wv path. If x and y do not lie on a common edge then they can be joined by two independent Wv paths, and it they do not lie on a common facet then they can be joined by three independent Wv paths. Results are obtained dealing with the question “when is a shortest path a Wv path?” Also using ideas related to Wv paths it is shown that any two vertices of a polytope with n k-dimensional faces can be joined by a path of length at most (3k−3)n