DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

IMPACT FACTOR 2018: 0.741

SCImago Journal Rank (SJR) 2018: 0.763

Rejection Rate (2017-2018): c. 84%

Discussiones Mathematicae Graph Theory

PDF

Discussiones Mathematicae Graph Theory 16(2) (1996) 207-217
DOI: 10.7151/dmgt.1035

ON LIGHT SUBGRAPHS IN PLANE GRAPHS OF MINIMUM DEGREE FIVE

Stanislav Jendrol'

Department of Geometry and Algebra, P.J. Safárik University, Jesenná 5
and Institute of Mathematics, Slovak Academy of Sciences
041 54 Košice, Slovak Republic

email: jendrol@Košice.upjs.sk

Tomáš Madaras

Department of Geometry and Algebra, P.J. Šafárik University
Jesenná 5, 041 54 Košice, Slovak Republic

email:
madaras@duro.upjs.sk

Abstract

A subgraph of a plane graph is light if the sum of the degrees of the vertices of the subgraph in the graph is small. It is well known that a plane graph of minimum degree five contains light edges and light triangles. In this paper we show that every plane graph of minimum degree five contains also light stars K1,3 and K1,4 and a light 4-path P4. The results obtained for K1,3 and P4 are best possible.

Keywords: planar graph, light subgraph, star, triangulation.

1991 Mathematics Subject Classification: 05C75, 05C10, 52B10.

References

[1] J.A. Bondy and U.S.R. Murty, Graph theory with applications (North Holland, Amsterdam 1976).
[2] O.V. Borodin, Solution of problems of Kotzig and Grünbaum concerning the isolation of cycles in planar graphs, Math. Notes 46 (1989) 835-837, doi: 10.1007/BF01139613.
[3] O.V. Borodin and D.P. Sanders, On light edges and triangles in planar graphs of minimum degree five, Math. Nachr. 170 (1994) 19-24, doi: 10.1002/mana.19941700103.
[4] I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs and Combinatorics (to appear).
[5] P. Franklin, The four colour problem, Amer. J. Math. 44 (1922) 225-236; or in: N.L. Biggs, E.K. Lloyd, R.J. Wilson (eds.), Graph Theory 1737 - 1936 (Clarendon Press, Oxford 1977).
[6] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat. źas. SAV (Math. Slovaca) 5 (1955) 111-113.
[7] A. Kotzig, Extremal polyhedral graphs, Ann. New York Acad. Sci. 319 (1979) 569-570.
[8] P. Wernicke, Über den kartographischen Vierfarbensatz, Math. Ann. 58 (1904) 413-426, doi: 10.1007/BF01444968.