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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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