Geodesic
Note on the weight of paths in plane triangulations of minimum degree 4 and 5
Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 173-180

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The weight of a path in a graph is defined to be the sum of degrees of its vertices in entire graph. It is proved that each plane triangulation of minimum degree 5 contains a path P₅ on 5 vertices of weight at most 29, the bound being precise, and each plane triangulation of minimum degree 4 contains a path P₄ on 4 vertices of weight at most 31.
Keywords: weight of path, plane graph, triangulation 
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Madaras, Tomás. Note on the weight of paths in plane triangulations of minimum degree 4 and 5. Discussiones Mathematicae. Graph Theory, Tome 20 (2000) no. 2, pp. 173-180. http://geodesic.mathdoc.fr/item/DMGT_2000_20_2_a1/