Annals of Combinatorics 2(1998) 313-324Light Paths with an Odd Number of Vertices in Large Polyhedral Maps S. Jendrol and H. -J. Voss Department of Geometry and Algebra, P.J. Šafárik University and Institute of Mathematics, Slovak Academy of Sciences, Jesenná 5, 041 54 Koš sice, Slovakia Department of Algebra, Technical University Dresden, Mommsenstrasse 13, D--01062 Dresden, Germany Received March 12, 1998 AMS Subject Classification: 05Cl0, 05C38, 52B10 Abstract. Let P_{k} be a path on k vertices. In this paper we prove that (1) every polyhedral map on the torus and the Klein bottle contains a path P_{k} such that each of its vertices has degree ≤ 6k-2 if k is odd, k≥ 3, (2) every large polyhedral map on any compact 2-manifold M with Euler characteristic Χ (M)<0 contains a path P_{k} such that each of its vertices has degree ≤ 6k-2 if k is odd, k≥ 3, (3) moreover, these bounds are attained. For k=1 or k even, k≥ 2, the bound is 6k which has been proved in our previous paper. Keywords: path, polyhedral map, embeddings |