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.
Maximum vertex and face degree of oblique graphs
Let G=(V,E,F) be a 3-connected simple graph imbedded into a surface S with vertex set V, edge set E and face set F. A face @a is an -face if @a is a k-gon and the degrees of the vertices incident with @a in the cyclic order are a"1,a"2,...,a"k. The lexicographic minimum such that @a is a -face is called the type of @a. Let z be an integer. We consider z-oblique graphs, i.e. such graphs that the number of faces of each type is at most z. We show an upper bound for the maximum vertex degree of any z-oblique graph imbedded into a given surface. Moreover, an upper bound for the maximum face degree is presented. We also show that there are only finitely many oblique graphs imbedded into non-orientable surfaces.
Shortness parameters of families of regular planar graphs with two or three types of face
This paper is concerned with non-Hamiltonian planar graphs. It is shown that the class of 3-connected cubic planar graphs whose faces are all pentagons or 10-gons contains non-Hamiltonian members and that the shortness coefficient of this class of graphs is less than unity. For several classes of 3-connected regular planar graphs, of valency 4 or 5, it is shown that the shortness exponent is less than unity.