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 α is an 〈 a 1 , a 2 , … , a k 〉 -face if α is a k -gon and the degrees of the vertices incident with α in the cyclic order are a 1 , a 2 , … , a k . The lexicographic minimum 〈 b 1 , b 2 , … , b k 〉 such that α is a 〈 b 1 , b 2 , … , b k 〉 -face is called the t y p e of α . 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.