Light paths with an odd number of vertices in polyhedral maps

Abstract

Let P _k be a path on k vertices. In an earlier paper we have proved that each polyhedral map G on any compact 2-manifold $M$ with Euler characteristic $x\left( M \right) \leqslant {\text{0}}$ contains a path P _k such that each vertex of this path has, in G, degree $\leqslant k\left[ {\frac{{5 + \sqrt {49 - 24 \times \left( M \right)} }}{2}} \right]$. Moreover, this bound is attained for k = 1 or k ≥ 2, k even. In this paper we prove that for each odd $k \geqslant \frac{{\text{4}}}{{\text{3}}}\left[ {\frac{{5 + \sqrt {49 - 24 \times \left( M \right)} }}{2}} \right] + 1$, this bound is the best possible on infinitely many compact 2-manifolds, but on infinitely many other compact 2-manifolds the upper bound can be lowered to $\left[ {\left( {k - \frac{{\text{1}}}{{\text{3}}}} \right)\frac{{5 + \sqrt {49 - 24 \times \left( M \right)} }}{2}} \right]$.