Abstract
In this paper it is proved that every 3-connected planar graph contains a path on 3 vertices each of which is of degree at most 15 and a path on 4 vertices each of which has degree at most 23. Analogous results are stated for 3-connected planar graphs of minimum degree 4 and 5. Moreover, for every pair of integers n ≥ 3, k ≥ 4 there is a 2-connected planar graph such that every path on n vertices in it has a vertex of degree k.
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Jendroľ, S. Paths with restricted degrees of their vertices in planar graphs. Czechoslovak Mathematical Journal 49, 481–490 (1999). https://doi.org/10.1023/A:1022411100562
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DOI: https://doi.org/10.1023/A:1022411100562