Low edges in 3-polytopes

On polytopes simple in edges

In this paper, we will study the h-vectors of a slightly more general class of polytopes. A d-polytope is called simple in edges if each its edge is incident exactly to d − 1 facets. We will prove that for any polytope simple in edges all numbers h[d/2], h[d/2]+1,. . ., hd are nonnegative and hk 6 hd−k for k 6 d/2. Polytopes simple in edges appear for instance as (closures of) fundamental polyh...

An inequality concerning edges of minor weight in convex 3-polytopes

Let eij be the number of edges in a convex 3–polytope joining the vertices of degree i with the vertices of degree j. We prove that for every convex 3-polytope there is 20e3,3 +25e3,4 +16e3,5 +10e3,6 + 6 2 3 e3,7+5e3,8+2 1 2 e3,9+2e3,10+16 2 3 e4,4+11e4,5+5e4,6+1 2 3 e4,7+5 1 3 e5,5+ 2e5,6 ≥ 120; moreover, each coefficient is the best possible. This result brings a final answer to the conjectur...

Regular and chiral polytopes in low dimensions

There are two main thrusts in the theory of regular and chiral polytopes: the abstract, purely combinatorial aspect, and the geometric one of realizations. This brief survey concentrates on the latter. The dimension of a faithful realization of a finite abstract regular polytope in some euclidean space is no smaller than its rank, while that of a chiral polytope must strictly exceed the rank. T...

Geometric shellings of 3-polytopes

A total order of the facets of a polytope is a geometric shelling if there exists a combinatorially equivalent polytope in which the corresponding order of facets becomes a line shelling. The subject of this paper is (geometric) shellings of 3-polytopes. Recently, a graph theoretical characterization of geometric shellings of 3-polytopes were given by Holt & Klee and Mihalisin & Klee. We rst gi...

Cyclic coloration of 3-polytopes