Light paths with an odd number of vertices in large polyhedral maps

LetP_k be a path onk vertices. In this paper we prove that (1) every polyhedral map on the torus and the Klein bottle contains a pathP_k such that each of its vertices has degree 6k–2 ifk is odd,k3, (2) every large polyhedral map on any compact 2-manifoldM with Euler characteristic (M)<0 contains a pathP_k such that each of its vertices has degree 6k – 2 ifk is odd,k3, (3) moreover, these bounds are attained. Fork=1 ork even,k2, the bound is 6k which has been proved in our previous paper.