Polyhedral graphs with restricted number of faces of the same type Original Research Article

Let G=G(V,E,F) be a polyhedral graph with vertex set V, edge set E and face set F. A face α is an 〈a1,…,al〉-face if α is an l-gon and the degrees d(xi) of the vertices x1,…,xl incident with α in the cyclic order are a1,…,al, respectively. The lexicographic minimum 〈b1,…,bl〉 such that α is a 〈b1,…,bl〉-face is called the type of α. Furthermore let z be a given integer. We consider polyhedral graphs where the number of faces of each type is restricted by z. We prove that there is only a finite number of such graphs.