On 3-connected plane graphs without triangular faces

We prove that each polyhedral triangular face free map G on a compact 2-dimensional manifold Rd with Euler characteristic chi ( M) contains a k-path, i.e., a path on k vertices, such that each vertex of this path has, in G, degree at most (5/2) k if M is a sphere S-0 and at most (k/2)[(5 + root 49 - 24 chi(M))/2] if M not equal S-0 or does not contain any k-path. We show that For even k this bound is best possible. Moreover, we show that for any graph other than a path no similiar estimation exists. (C) 1999 Academic Press.