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On $$3$$-connected plane graphs without triangular faces. (English) Zbl 1027.05030
Summary: We prove that each polyhedral triangular face-free map $$G$$ on a compact 2-dimensional manifold $$\mathbb{M}$$ with Euler characteristics $$\chi(\mathbb{M})$$ contains a $$k$$-path, i.e., a path on $$k$$ vertices, such that each vertex of this path has, in $$G$$, degree at most $$(5/2)k$$ if $$\mathbb{M}$$ is a sphere $$\mathbb{S}_0$$ and at most $$(k/2)\lfloor(5+ \sqrt{49-24\chi(\mathbb{M})})/2\rfloor$$ if $$\mathbb{M}\neq \mathbb{S}_0$$ or does not contain any $$k$$-path. We show that for even $$k$$ this bound is best possible. Moreover, we show that for any graph other than a path no similar estimation exists.

MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles
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