## Light paths with an odd number of vertices in polyhedral maps.(English)Zbl 1079.05502

Summary: 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 $$\mathbb M$$ with Euler characteristic $$\chi (\mathbb M)\leq 0$$ contains a path $$P_k$$ such that each vertex of this path has, in $$G$$, degree $$\leq k\lfloor A \rfloor$$ with $$2A = 5+ \sqrt {49-24\chi (\mathbb M)}$$. Moreover, this bound is attained for $$k=1$$ or $$k\geq 2$$, $$k$$ even. In this paper we prove that for each odd $$k\geq \frac 43 \lfloor A\rfloor +1$$, 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 $$\lfloor (k-\frac 13)A\rfloor$$.

### MSC:

 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles 52B10 Three-dimensional polytopes

### Keywords:

graphs; path; polyhedral map; embeddings
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### References:

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