Convex 3-polytopes with exactly two types of edges
Abstract We consider convex 3-polytopes with exactly two types of edges. The questions of the existence of such 3-polytopes are solved. The cardinalities of all classes are determined.
Non-Hamiltonian Simple 3-Polytopes with Only One Type of Face Besides Triangles
It is shown, for q = 8, 9 and 10, that the class of simple 3-polytopes whose faces are of only two types, triangles and q -gons, contains non-Hamiltonian members and even has shortness coefficient less than unity.
Non-Hamiltonian simple 3-polytopes having just two types of faces
It is shown that for every value of an integer k, k>=11, there exist 3-valent 3-connected planar graphs having just two types of faces, pentagons and k-gons, and which are non- Hamiltonian. Moreover, there exist @[email protected](k) > 0, for these values of k, and sequences (G"n)^~"n"="1 of the said graphs for which V(G"n)->~ and the size of a largest circuit of G"n is at most ([email protected])V(G"n); similar result for the size of a largest path in such graphs is established for all k, k>=11, except possibly for k = 14, 17, 22 and k = 5m+ 5 for all m>=2.