On 3-Connected Plane Graphs without Triangular Faces

We prove that each polyhedral triangular face free map G on a compact 2-dimensional manifold M with Euler characteristic χ(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 S0 and at most (k/2)⌊(5+49−24χ(M))/2⌋ if M≠S0 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 similar estimation exists.