Abstract
A graph G is k-degenerated if it can be deleted by subsequent removals of vertices of degree k or less. We survey known results on the size of maximal k- degenerated induced subgraph in a planar graph. In addition, we sketch the proof that every planar graph of order n has a 4-degenerated induced subgraph of order at least 8/9 · n. We also show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.
The first author acknowledges partial support from the research grant 7/TU/13 and from the APW grant ESF-EC-0009-10 within the EUROCORES Programme EUROGIGA (project GReGAS) of the European Science Foundation.
The work of the second author leading to this invention has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC grant agreement no. 259385.
The third author acknowledges partial support from the research grants NSF No. 11171730 and ZJNSF No. Z6110786.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
M. O. Albertson and D. M. Berman, A conjecture on planar graphs, In: “Graph Theory and Related Topics” J. A. Bondy and U. S. R. Murty, (eds.), Academic Press, 1979, 357.
N. Alon, J. Kahn and P. D. Seymour, Large induced degenerate subgraphs, Graphs and Combinatorics 3 (1987), 203–211.
J. Barát, G. Joret and D. R. Wood, Disproof of the list Hadwiger Conjecture, Electron. J. Combin. 18 (2011), #P232.
O. V. Borodin, On acyclic colorings of planar graphs, Discrete Math. 25 (1979), 211–236.
C. Thomassen, Every planar graph is 5-choosable, Journal of Comb. Theory Ser B 62 (1994), 180–181.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2013 Scuola Normale Superiore Pisa
About this paper
Cite this paper
Lukot’ka, R., Mazák, J., Zhu, X. (2013). Degenerated induced subgraphs of planar graphs. In: Nešetřil, J., Pellegrini, M. (eds) The Seventh European Conference on Combinatorics, Graph Theory and Applications. CRM Series, vol 16. Edizioni della Normale, Pisa. https://doi.org/10.1007/978-88-7642-475-5_42
Download citation
DOI: https://doi.org/10.1007/978-88-7642-475-5_42
Publisher Name: Edizioni della Normale, Pisa
Print ISBN: 978-88-7642-474-8
Online ISBN: 978-88-7642-475-5
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)