On approximating the achromatic number

Guy Kortsarz, Robert Krauthgamer

Research output: Contribution to journalArticlepeer-review


The achromatic number problem is to legally color the vertices of an input graph with the maximum number of colors, denoted ψ* , so that every two color classes share at least one edge. This problem is known to be NP-hard. For general graphs we give an algorithm that approximates the achromatic number within a ratio of O(n - log log n/ log n). This improves over the previously known approximation ratio of O(n/ √log n), due to Chaudhary and Vishwanathan [Proceedings of the Eighth Annual ACM-SIAM Symposium on Discrete Algorithms, New Orleans, LA, 1997, pp. 558-563]. For graphs of girth at least 5 we give an algorithm with an approximation ratio O(min{n1/3, √ψ*}). This improves over an approximation ratio O(√ψ*) = O(n3/8) for the more restricted case of graphs with girth at least 6, due to Krysta and Loryś [Proceedings of the Seventh Annual European Symposium on Algorithms, Lecture Notes in Comput. Sci. 1643, Springer-Verlag, Berlin, 1999, pp. 402-413]. We also give the first hardness result for approximating the achromatic number. We show that for every fixed ε > 0 there is no 2 - ε approximation algorithm, unless P = NP.

Original languageEnglish
Pages (from-to)408-422
Number of pages15
JournalSIAM Journal on Discrete Mathematics
Issue number3
StatePublished - May 2001


  • Achromatic number
  • Approximation algorithms
  • Graph coloring


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