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Subgraphs with restricted degrees of their vertices in planar 3-connected graphs. (English) Zbl 0891.05025
Every 3-connected planar graph $$G$$ either contains a path with $$k$$ vertices each of degree at most $$5k$$ or does not contain any path with $$k$$ vertices; the bound $$5k$$ is the best possible. Moreover, for every connected planar graph $$H$$ other than a path and for every integer $$m\geq 3$$ there is a 3-connected planar graph $$G$$ such that each copy of $$H$$ in $$G$$ contains a vertex of degree at least $$m$$.

##### MSC:
 05C10 Planar graphs; geometric and topological aspects of graph theory 05C38 Paths and cycles
##### Keywords:
planar graph; path; degree
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##### References:
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