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 alpha is an (a(1), a(2),..,a(k))-face if alpha is a k-gon and the degrees of the vertices incident with alpha in the cyclic order are a(1), a(2),..., a(k). The lexicographic minimum (b(1), b(2),...,b(k)) such that alpha is a (b(1), b(2),...,b(k))-face is called the type of alpha. 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. (C) 2008 Elsevier B.V. All rights reserved.