Abstract
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,k≥3, (2) every large polyhedral map on any compact 2-manifoldM with Euler characteristic χ(M)<0 contains a pathP k such that each of its vertices has degree ≤ 6k − 2 ifk is odd,k≥3, (3) moreover, these bounds are attained. Fork=1 ork even,k≥2, the bound is 6k which has been proved in our previous paper.
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Jendrol', S., Voss, H.J. Light paths with an odd number of vertices in large polyhedral maps. Annals of Combinatorics 2, 313–324 (1998). https://doi.org/10.1007/BF01608528
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DOI: https://doi.org/10.1007/BF01608528