On light subgraphs in plane graphs of minimum degree five

Stanislav Jendrol'; Tomáš Madaras

Discussiones Mathematicae Graph Theory (1996)

  • Volume: 16, Issue: 2, page 207-217
  • ISSN: 2083-5892

Abstract

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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 K 1 , 3 and K 1 , 4 and a light 4-path P₄. The results obtained for K 1 , 3 and P₄ are best possible.

How to cite

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Stanislav Jendrol', and Tomáš Madaras. "On light subgraphs in plane graphs of minimum degree five." Discussiones Mathematicae Graph Theory 16.2 (1996): 207-217. <http://eudml.org/doc/270171>.

@article{StanislavJendrol1996,
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 $K_\{1,3\}$ and $K_\{1,4\}$ and a light 4-path P₄. The results obtained for $K_\{1,3\}$ and P₄ are best possible.},
author = {Stanislav Jendrol', Tomáš Madaras},
journal = {Discussiones Mathematicae Graph Theory},
keywords = {planar graph; light subgraph; star; triangulation; plane graph; subgraph; stars},
language = {eng},
number = {2},
pages = {207-217},
title = {On light subgraphs in plane graphs of minimum degree five},
url = {http://eudml.org/doc/270171},
volume = {16},
year = {1996},
}

TY - JOUR
AU - Stanislav Jendrol'
AU - Tomáš Madaras
TI - On light subgraphs in plane graphs of minimum degree five
JO - Discussiones Mathematicae Graph Theory
PY - 1996
VL - 16
IS - 2
SP - 207
EP - 217
AB - 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 $K_{1,3}$ and $K_{1,4}$ and a light 4-path P₄. The results obtained for $K_{1,3}$ and P₄ are best possible.
LA - eng
KW - planar graph; light subgraph; star; triangulation; plane graph; subgraph; stars
UR - http://eudml.org/doc/270171
ER -

References

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  1. [1] J.A. Bondy and U.S.R. Murty, Graph theory with applications (North Holland, Amsterdam 1976). Zbl1226.05083
  2. [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. Zbl0717.05034
  3. [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. Zbl0813.05020
  4. [4] I. Fabrici and S. Jendrol', Subgraphs with restricted degrees of their vertices in planar 3-connected graphs, Graphs and Combinatorics (to appear). Zbl0891.05025
  5. [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). Zbl48.0664.02
  6. [6] A. Kotzig, Contribution to the theory of Eulerian polyhedra, Mat. źas. SAV (Math. Slovaca) 5 (1955) 111-113. 
  7. [7] A. Kotzig, Extremal polyhedral graphs, Ann. New York Acad. Sci. 319 (1979) 569-570. 
  8. [8] P. Wernicke, Über den kartographischen Vierfarbensatz, Math. Ann. 58 (1904) 413-426, doi: 10.1007/BF01444968. Zbl35.0511.01

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