Geodesic
Paths with restricted degrees of their vertices in planar graphs
Czechoslovak Mathematical Journal, Tome 49 (1999) no. 3, pp. 481-490.

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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 $n\ge 3$, $ k\ge 4$ there is a $2$-connected planar graph such that every path on $n$ vertices in it has a vertex of degree $k$.
Classification : 05C35, 05C38 
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     author = {Jendro\v{l}, Stanislav},
     title = {Paths with restricted degrees of their vertices in planar graphs},
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     pages = {481--490},
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Jendroľ, Stanislav. Paths with restricted degrees of their vertices in planar graphs. Czechoslovak Mathematical Journal, Tome 49 (1999) no. 3, pp. 481-490. http://geodesic.mathdoc.fr/item/CMJ_1999__49_3_a2/