Abstract
The achromatic number for a graph G=〈V,E〉 is the largest integer m such that there is a partition of V into disjoint independent sets {V1,…,Vm} such that for each pair of distinct sets Vi, Vj, Vi∪Vj is not an independent set in G. Yannakakis and Gavril (1980, SIAM J. Appl. Math.38, 364–372) proved that determining this value for general graphs is NP-complete. For n-vertex graphs we present the first o(n) approximation algorithm for this problem. We also present an O(n5/12) approximation algorithm for graphs with girth at least 5 and a constant approximation algorithm for trees.
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