Skip to main content
SpringerLink
Log in

Graph Subcolorings: Complexity and Algorithms

(Extended Abstract)

  • Conference paper
  • First Online:
Graph-Theoretic Concepts in Computer Science (WG 2001)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2204))

Included in the following conference series:

Abstract

In a graph coloring, each color class induces a disjoint union of isolated vertices. A graph subcoloring generalizes this concept, since here each color class induces a disjoint union of complete graphs. Erdős and independently Albertson et al. proved that every graph of maximum degree at most 3 has a 2-subcoloring.We point out in this paper that this fact is best possible with respect to degree-constraints by showing that the problem of recognizing 2-subcolorable graphs with maximum degree 4 is NP-complete, even when restricted to triangle-free planar graphs. Moreover, in general, for fixed k, recognizing k-subcolorable graphs is NP-complete on graphs with maximum degree at most k 2. In contrast, we show that, for arbitrary k, k-SUBCOLORABILITY can be computed efficiently on graphs of bounded treewidth and on cographs.

Research supported in part by EU ARACNE project HPRN-CT-1999-00112 and EU APPOL project IST-1999-14084

Financial support by Deutsche Forschungsgemeinschaft is gratefully acknowledged.

Supported by the Ministery of Education of the Czech Republic as project LN00A056

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. D. Achlioptas, The complexity of G-free graph colourability, Discrete Math, 165/166 (1997), 31–38.

    Article  MathSciNet  Google Scholar 

  2. M.O. Albertson, R.E. Jamison, S.T. Hedetniemi, S.C. Locke, The subchromatic number of a graph, Discrete Math., 74 (1989), 33–49.

    Article  MATH  MathSciNet  Google Scholar 

  3. H.L. Bodlaender, A linear time algorithm for finding tree-decompositions of small treewidth. SIAM J. Computing, 25 (1996), 1305–1317.

    Article  MATH  MathSciNet  Google Scholar 

  4. I. Broere, C.M. Mynhardt, Generalized colorings of outerplanar and planar graphs, Proc. Graph Theory with Applications to Algorithms and Computer Science, Kalamazoo, Mich., (1984), 151–161.

    Google Scholar 

  5. J.L. Brown, D.G. Corneil, On generalized graph colorings, J. Graph Theory, 11 (1987), 87–99.

    Article  MATH  MathSciNet  Google Scholar 

  6. D.G. Corneil, H. Lerchs, L. Stewart Burlingham, Complement reducible graphs. Discrete Appl. Math., 3 (1981), 163–174.

    Article  MATH  MathSciNet  Google Scholar 

  7. D.G. Corneil, Y. Perl, L.K. Stewart, A linear recognition algorithm for cographs, SIAM J. Computing, 14 (1985), 926–934.

    Article  MATH  MathSciNet  Google Scholar 

  8. B. Courcelle, Graph rewriting: an algebraic and logical approach, Handbook of Theoretical Computer Science, volume B, (1990), 192–242.

    Google Scholar 

  9. B. Courcelle, The monadic second-order logic of graphs I: Recognizable sets of finite graphs, Information and Computation, 85 (1991), 12–75.

    Article  MathSciNet  Google Scholar 

  10. P. Erdős, Bipartite subgraphs of graphs, Math. Lapok, 18 (1967), 283–288.

    Google Scholar 

  11. M. Garey, D.S. Johnson, Computers and Intractability: A Guide to the Theory of NP-completeness, W.H. Freeman, San Fransisco, 1979.

    MATH  Google Scholar 

  12. M.R. Garey, D.S. Johnson, L. Stockmayer, Some simplified NP-complete graph problems, Theoretical Computer Science, 1 (1976), 237–267.

    Article  MATH  MathSciNet  Google Scholar 

  13. T. Kloks, Treewidth — Computations and Approximations, Lecture Notes in Computer Science no. 842, Springer-Verlag, 1994.

    MATH  Google Scholar 

  14. F. Mafray, M. Preissman, On the NP-completeness of k-colorability problem of triangle-free graphs, Discrete Math., 162 (1996), 313–317.

    Article  MathSciNet  Google Scholar 

  15. C.M. Mynhardt, I. Broere, Generalized colorings of graphs, Proc. Graph Theory with Applications to Algorithms and Computer Science, Kalamazoo, Mich., (1984), 583–594.

    Google Scholar 

  16. N. Robertson and P.D. Seymour, Graph minors. II. Algorithmic aspects of treewidth, Journal of Algorithms, 7 (1986), 309–322.

    Article  MATH  MathSciNet  Google Scholar 

  17. T.J. Schaefer, The complexity of the satisfability problem, Proc. 10th Ann. ACM Symp. on theory of computing (1978), 216–226.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Informatik und Praktische Mathematik, Christian-Albrechts-Universität zu Kiel, 24098, Kiel, Germany

    Jiří Fiala, Klaus Jansen & Eike Seidel

  2. Institute of Theoretical Computer Science, Charles University, Prague, Czech Republic

    Jiří Fiala

  3. Institut für Theoretische Informatik, Fachbereich Informatik, Universität Rostock, 18051, Rostock, Germany

    Van Bang Le

Authors
  1. Jiří Fiala
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Klaus Jansen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Van Bang Le
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Eike Seidel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, University of Rostock, 18051, Rostock, Germany

    Andreas Brandstädt  & Van Bang Le  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Fiala, J., Jansen, K., Le, V.B., Seidel, E. (2001). Graph Subcolorings: Complexity and Algorithms. In: Brandstädt, A., Le, V.B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2001. Lecture Notes in Computer Science, vol 2204. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45477-2_15

Download citation

  • DOI: https://doi.org/10.1007/3-540-45477-2_15

  • Published:

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42707-0

  • Online ISBN: 978-3-540-45477-9

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation