Discussiones
Mathematicae Graph Theory 18(2) (1998) 159-164
DOI: https://doi.org/10.7151/dmgt.1071
SHORT CYCLES OF LOW WEIGHT IN NORMAL PLANE MAPS WITH MINIMUM DEGREE 5
Oleg V. Borodin Novosibirsk State University |
Douglas R. Woodall Department of Mathematics, University of Nottingham |
Abstract
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(K1,4) ≤ 30. These results answer a question posed by Kotzig in 1979 and recent questions of Jendrol' and Madaras.
Keywords: planar graphs, plane triangulation.
1991 Mathematics Subject Classification: 05C75, 05C10, 05C38.
References
[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. |
Received 29 August 1997
Revised 25 March 1998
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