Geodesic
Short cycles of low weight in normal plane maps with minimum degree 5
Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 2, pp. 159-164.

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In this note, precise upper bounds are determined for the minimum degree-sum of the vertices of a 4-cycle and a 5-cycle in a plane triangulation with minimum degree 5: w(C₄) ≤ 25 and w(C₅) ≤ 30. These hold because a normal plane map with minimum degree 5 must contain a 4-star with w(K_1,4) ≤ 30. These results answer a question posed by Kotzig in 1979 and recent questions of Jendrol' and Madaras.
Keywords: planar graphs, plane triangulation 
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Borodin, Oleg; Woodall, Douglas. Short cycles of low weight in normal plane maps with minimum degree 5. Discussiones Mathematicae. Graph Theory, Tome 18 (1998) no. 2, pp. 159-164. http://geodesic.mathdoc.fr/item/DMGT_1998_18_2_a1/

[1] O.V. Borodin, Solution of Kotzig's and Grünbaum's problems on the separability of a cycle in a planar graph, Matem. Zametki 46 (5) (1989) 9-12. (in Russian)

[2] O.V. Borodin and D.R. Woodall, Vertices of degree 5 in plane triangulations (manuscript, 1994).

[3] S. Jendrol' and T. Madaras, On light subgraphs in plane graphs of minimal degree five, Discussiones Math. Graph Theory 16 (1996) 207-217, doi: 10.7151/dmgt.1035.

[4] A. Kotzig, From the theory of eulerian polyhedra, Mat. Cas. 13 (1963) 20-34. (in Russian)

[5] A. Kotzig, Extremal polyhedral graphs, Ann. New York Acad. Sci. 319 (1979) 569-570.