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A local property of large polyhedral maps on compact 2-dimensional manifolds. (English) Zbl 0933.05044
Summary: We prove that each polyhedral map $$G$$ on a compact 2-manifold, which has large enough vertices, contains a $$k$$-path, a path on $$k$$ vertices, such that each vertex of it has, in $$G$$, degree at most $$6k$$; this bound being best possible for $$k$$ even. Moreover, if $$G$$ has large enough vertices of degree $$>6k$$, than it contains a $$k$$-path such that each of its vertices has degree, in $$G$$, at most $$5k$$; this bound is best possible for any $$k$$.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C35 Extremal problems in graph theory 05C38 Paths and cycles 54C25 Embedding
##### Keywords:
light graph; embeddings; polyhedral map; compact 2-manifold; path; bound
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