An improved approximation of the achromatic number on bipartite graphs

The achromatic number of a graph G = (V, E) with |V| = n vertices is the largest number k with the following property: the vertices of G can be partitioned into k independent subsets {V_i}_1≤i≤k such that for every distinct pair of subsets V_i, V_j in the partition, there is at least one edge in E that connects these subsets. We describe a greedy algorithm that computes the achromatic number of a bipartite graph within a factor of O(n^4/5) of the optimal. Prior to our work, the best known approximation factor for this problem was n log log n/ log n as shown by Kortsarz and Krauthgamer [SIAM J. Discrete Math., 14 (2001), pp. 408-422].