Geodesic
Minimal vertex degree sum of a 3-path in plane maps
Discussiones Mathematicae. Graph Theory, Tome 17 (1997) no. 2, pp. 279-284

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Let wₖ be the minimum degree sum of a path on k vertices in a graph. We prove for normal plane maps that: (1) if w₂ = 6, then w₃ may be arbitrarily big, (2) if w₂ 6, then either w₃ ≤ 18 or there is a ≤ 15-vertex adjacent to two 3-vertices, and (3) if w₂ 7, then w₃ ≤ 17.
Keywords: planar graph, structure, degree, path, weight 
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     author = {Borodin, O.},
     title = {Minimal vertex degree sum of a 3-path in plane maps},
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Borodin, O. Minimal vertex degree sum of a 3-path in plane maps. Discussiones Mathematicae. Graph Theory, Tome 17 (1997) no. 2, pp. 279-284. http://geodesic.mathdoc.fr/item/DMGT_1997_17_2_a6/