Efficient Approximation Algorithms for the Achromatic Number

The achromatic number problem is as follows: given a graph G = (V,E), find the greatest number of colors in a coloring of the vertices of G such that adjacent vertices get distinct colors and for every pair of colors some vertex of the first color and some vertex of the second color are adjacent. This problem is NP-complete even for trees. We present improved polynomial time approximation algorithms for the problem on graphs with large girth and for trees, and linear time approximation algorithms for trees with bounded maximum degree. We also improve the lower bound of Farber et al. for the achromatic number of trees with maximum degree bounded by three.