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Annual Symposium on Theoretical Aspects of Computer Science

STACS 2000: STACS 2000 pp 395–406Cite as

λ-Coloring of Graphs

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 1770))

Abstract

A λ-coloring of a graph G is an assignment of colors from the set 0, ..., λ to the vertices of a graph G such that vertices at distance at most two get different colors and adjacent vertices get colors which are at least two apart. The problem of finding λ-colorings with small or optimal λ arises in the context of radio frequency assignment. We show that the problems of finding the minimum λ for planar graphs, bipartite graphs, chordal graphs and split graphs are NP-Complete. We then give approximation algorithms for λ-coloring and compute upperbounds of the best possible λ for outerplanar graphs, planar graphs, graphs of treewidth k, permutation and split graphs. With the exception of the split graphs, all the above bounds for λ are linear in Δ, the maximum degree of the graph. For split graphs, we give a bound of λ ≤ Δ 1.5+2Δ+2 and show that there are split graphs with λ = Ω(Δ 1.5). Similar results are also given for variations of the λ-coloring problem.

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Author information

Authors and Affiliations

  1. Department of Computer Science, Utrecht University, Padualaan 14, 3584 CH, Utrecht, The Netherlands

    Hans L. Bodlaender, Richard B. Tan & Jan van Leeuwen

  2. Department of Mathematics and Computer Science, Vrije Universiteit, 1081 HV, Amsterdam, The Netherlands

    Ton Kloks

  3. Department of Computer Science, University of Sciences & Arts of Oklahoma, Chickasha, OK, 73018, USA

    Richard B. Tan

Authors
  1. Hans L. Bodlaender
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  2. Ton Kloks
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  3. Richard B. Tan
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  4. Jan van Leeuwen
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Editor information

Editors and Affiliations

  1. Faculty of Computer Science, Dept. of Theoretical Computer Science, Universität Dresden, 01062, Dresden, Germany

    Horst Reichel

  2. Université de Lille, LIFL - UPRESA 8022, CNRS, Bât. M3 - UFR IEEA, 59655, Villeneuve d’ASCQ Cedex, France

    Sophie Tison

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© 2000 Springer-Verlag Berlin Heidelberg

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Bodlaender, H.L., Kloks, T., Tan, R.B., van Leeuwen, J. (2000). λ-Coloring of Graphs. In: Reichel, H., Tison, S. (eds) STACS 2000. STACS 2000. Lecture Notes in Computer Science, vol 1770. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-46541-3_33

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  • DOI: https://doi.org/10.1007/3-540-46541-3_33

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  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-67141-1

  • Online ISBN: 978-3-540-46541-6

  • eBook Packages: Springer Book Archive

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