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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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/

[1] O.V. Borodin, Solution of Kotzig's and Grünbaum's problems on the separability of a cycle in plane graph, (in Russian), Matem. zametki 48 (6) (1989) 9-12.

[2] O.V. Borodin, Triangulated 3-polytopes without faces of low weight, submitted.

[3] H. Enomoto and K. Ota, Properties of 3-connected graphs, preprint (April 21, 1994).

[4] K. Ando, S. Iwasaki and A. Kaneko, Every 3-connected planar graph has a connected subgraph with small degree sum I, II (in Japanese), Annual Meeting of Mathematical Society of Japan, 1993.

[5] Ph. Franklin, The four colour problem, Amer. J. Math. 44 (1922) 225-236, doi: 10.2307/2370527.

[6] S. Jendrol', Paths with restricted degrees of their vertices in planar graphs, submitted.

[7] S. Jendrol', A structural property of 3-connected planar graphs, submitted.

[8] A. Kotzig, Contribution to the theory of Eulerian polyhedra, (in Russian), Mat. Čas. 5 (1955) 101-103.