if , then every -dimensional face of is also a simplex. In particular, the Kleetope of any three-dimensional polyhedron is a simplicial polyhedron, a polyhedron in which all facets are triangles. Kleetopes may be used to generate polyhedra that do not have any Hamiltonian cycles: any path through one of the vertices added in the Kleetope construction must go into and out of the vertex through its neighbors in the original polyhedron, and if there are more new vertices than original vertices then there are not enough neighbors to go around. In particular, the Goldner–Harary graph, the Kleetope of the triangular bipyramid, has six vertices added in the Kleetope construction and only five in the bipyramid from which it was formed, so it is non-Hamiltonian; it is the simplest possible non-Hamiltonian simplicial polyhedron. If a polyhedron with vertices is formed by repeating the Kleetope construction some number of times, starting from a tetrahedron, then its longest path has length ; that is, the shortness exponent of these graphs is , approximately 0.630930. The same technique shows that in any higher dimension , there exist simplicial polytopes with shortness exponent . Similarly, used the Kleetope construction to provide an infinite family of examples of simplicial polyhedra with an even number of vertices that have no perfect matching. Kleetopes also have some extreme properties related to their vertex degrees: if each edge in a planar graph is incident to at least seven other edges, then there must exist a vertex of degree at most five all but one of whose neighbors have degree 20 or more, and the Kleetope of the Kleetope of the icosahedron provides an example in which the high-degree vertices have degree exactly 20. Notes References . . See also the same journal 6(2):33 (1975) and 8:104-106 (1977). Reference from listing of Harary's publications. . . . . Polyhedra Polytopes