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Italian Conference on Theoretical Computer Science

ICTCS 2001: Theoretical Computer Science pp 285–292Cite as

Distance Constrained Labeling of Precolored Trees

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2202))

Abstract

Graph colorings with distance constraints are motivated by the frequency assignment problem. The so called λ(p,q)-labeling problem asks for coloring the vertices of a given graph with integers from the range {0, 1, ..., λ} so that labels of adjacent vertices differ by at least p and labels of vertices at distance 2 differ by at least q, where p, q are fixed integers and integer λ is part of the input. It is known that this problem is NP-complete for general graphs, even when λ is fixed, i.e., not part of the input, but polynomially solvable for trees for (p,q)=(2,1). It was conjectured that the general case is also polynomial for trees. We consider the precoloring extension version of the problem (i.e., when some vertices of the input tree are already precolored) and show that in this setting the cases q=1 and q > 1 behave di.erently: the problem is polynomial for q=1 and any p, and it is NP-complete for any p > q > 1.

The authors acknowledge support of joint Czech U.S. grants KONTAKT ME338 and NSF-INT-9802416 during visits of the first two authors to Eugene, OR, and of the third author to Prague.

Partially supported by EU ARACNE project HPRN-CT-1999-00112. Supported by the Ministery of Education of the Czech Republic as project LN00A056

Research partially supported by Czech Research grant GAUK 158/99. Supported by the Ministery of Education of the Czech Republic as project LN00A056

Supported in part by the grant NSF-ANI-9977524.

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References

  1. Battiti, R., A. A. Bertossi, and M. A. Bonuccelli, Assigning codes in wireless networks: Bounds and scaling properties. Wireless Networks 5, 3, (1999), pp. 195–209.

    Article  Google Scholar 

  2. Chang, G. J. and D. Kuo, The L(2, 1)-labeling problem on graphs, SIAM J. Disc. Math. 9, (1996), pp. 309–316.

    Article  MATH  MathSciNet  Google Scholar 

  3. Edmonds, J., Paths, trees and flowers, Can. J. Math. 17, (1965), pp. 449–467.

    MATH  MathSciNet  Google Scholar 

  4. Fiala, J., T. Kloks, and J. Kratochv’il, Fixed-parameter complexity of λ-colorings, In: Graph Theoretic Concepts in Computer Science, WG’99, LNCS 1665, (1999), pp. 350–363.

    Chapter  Google Scholar 

  5. Fotakis, D., G. Pantziou, G. Pentaris, and P. Spirakis, Frequency assignment in mobile and radio networks, In: Networks in distributed computing. DIMACS workshop, Ser. Discrete Math. Theor. Comput. Sci. 45, (1997), pp. 73–90.

    Google Scholar 

  6. Garey, M. R. and D. S. Johnson, Computers and Intractability, W. H. Freeman, 1979.

    Google Scholar 

  7. Griggs, J. R. and R. K. Yeh, Labelling graphs with a condition at distance 2, SIAM J. Disc. Math. 5, (1992), pp. 586–595.

    Article  MATH  MathSciNet  Google Scholar 

  8. Georges, J. P. and D. W. Mauro, On the size of graphs labeled with a condition at distance two, Journal of Graph Theory 22, (1996), pp. 47–57.

    Article  MATH  MathSciNet  Google Scholar 

  9. van den Heuvel, J., R. A. Leese, and M. A. Shepherd, Graph labeling and radio channel assignment, Journal of Graph Theory, 29, (1998), pp. 263–283.

    Article  MATH  MathSciNet  Google Scholar 

  10. Jonas, K., Graph colorings analogues with a condition at distance two: L(2, 1)-labelings and list λ-labelings, Ph.D. thesis, University of South Carolina, 1993.

    Google Scholar 

  11. Hujter, M. and Zs. Tuza, Precoloring extension III. Classes of perfect graphs, Combin. Probab. Comput. 5, (1996), pp. 35–56.

    Article  MATH  MathSciNet  Google Scholar 

  12. Kratochvíl, J. and A. Sebő, Coloring precolored perfect graphs, J. Graph. Th. 25, (1997), pp. 207–215.

    Article  MATH  Google Scholar 

  13. Liu, D. D.-F. and R. K. Yeh, On distance two labellings of graphs, ARS Combinatorica 47, (1997), pp. 13–22.

    MATH  MathSciNet  Google Scholar 

  14. Yeh, K.-Ch., Labeling graphs with a condition at distance two, Ph.D. Thesis, University of South Carolina, 1990.

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institute for Theoretical Computer Science and Department of Applied Mathematics, Charles University, Prague, Germany

    Jiři Fiala & Jan Kratochvíl

  2. Christian Albrechts University, Kiel, Germany

    Jiři Fiala

  3. Department of Computer and Information Science, University of Oregon, Eugene, USA

    Andrzej Proskurowski

Authors
  1. Jiři Fiala
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  2. Jan Kratochvíl
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  3. Andrzej Proskurowski
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© 2001 Springer-Verlag Berlin Heidelberg

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Fiala, J., Kratochvíl, J., Proskurowski, A. (2001). Distance Constrained Labeling of Precolored Trees. In: Theoretical Computer Science. ICTCS 2001. Lecture Notes in Computer Science, vol 2202. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45446-2_18

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  • DOI: https://doi.org/10.1007/3-540-45446-2_18

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-42672-1

  • Online ISBN: 978-3-540-45446-5

  • eBook Packages: Springer Book Archive

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