Abstract
We have proved that every 3-connected planar graph G either contains a path on k vertices each of which has degree at most 5k or does not contain any path on 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 ≥ 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.
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Fabrici, I., Jendrol, S. Subgraphs with Restricted Degrees of Their Vertices in Planar 3-Connected Graphs. Graphs and Combinatorics 13, 245–250 (1997). https://doi.org/10.1007/BF03353001
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DOI: https://doi.org/10.1007/BF03353001