## Abstract

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{n^{1/3}, √ψ*}). This improves over an approximation ratio O(√ψ*) = O(n^{3/8}) 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 language | English (US) |
---|---|

Pages (from-to) | 408-422 |

Number of pages | 15 |

Journal | SIAM Journal on Discrete Mathematics |

Volume | 14 |

Issue number | 3 |

DOIs | |

State | Published - May 1 2001 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Mathematics(all)

## Keywords

- Achromatic number
- Approximation algorithms
- Graph coloring