, Volume 59, Issue 2, pp 169–194 | Cite as

Exact Algorithms for L(2,1)-Labeling of Graphs

  • Frédéric Havet
  • Martin Klazar
  • Jan Kratochvíl
  • Dieter KratschEmail author
  • Mathieu Liedloff


The notion of distance constrained graph labelings, motivated by the Frequency Assignment Problem, reads as follows: A mapping from the vertex set of a graph G=(V,E) into an interval of integers {0,…,k} is an L(2,1)-labeling of G of span k if any two adjacent vertices are mapped onto integers that are at least 2 apart, and every two vertices with a common neighbor are mapped onto distinct integers. It is known that for any fixed k≥4, deciding the existence of such a labeling is an NP-complete problem. We present exact exponential time algorithms that are faster than the naive O *((k+1) n ) algorithm that would try all possible mappings. The improvement is best seen in the first NP-complete case of k=4, where the running time of our algorithm is O(1.3006 n ). Furthermore we show that dynamic programming can be used to establish an O(3.8730 n ) algorithm to compute an optimal L(2,1)-labeling.


Graph Algorithm Moderately exponential time algorithm L(2,1)-labeling 


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© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Frédéric Havet
    • 1
  • Martin Klazar
    • 2
  • Jan Kratochvíl
    • 2
  • Dieter Kratsch
    • 3
    Email author
  • Mathieu Liedloff
    • 4
  1. 1.Projet Mascotte I3S (CNRS & UNSA) and INRIAINRIA Sophia-AntipolisSophia-Antipolis CedexFrance
  2. 2.Department of Applied Mathematics and Institute for Theoretical Computer ScienceCharles UniversityPraha 1Czech Republic
  3. 3.Laboratoire d’Informatique Théorique et AppliquéeUniversité Paul Verlaine–MetzMetz Cedex 01France
  4. 4.Laboratoire d’Informatique Fondamentale d’OrléansUniversité d’OrléansOrléans Cedex 2France

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