We prove that each polyhedral triangular face free map G on a compact 2-dim
ensional 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, d
egree 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 t
hat 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 Pr
ess.