Complexity Theory
Finding ψ(G) is an optimization problem. The decision problem for complete coloring can be phrased as:
- INSTANCE: a graph and positive integer
- QUESTION: does there exist a partition of into or more disjoint sets such that each is an independent set for and such that for each pair of distinct sets is not an independent set.
Determining the achromatic number is NP-hard; determining if it is greater than a given number is NP-complete, as shown by Yannakakis and Gavril in 1978 by transformation from the minimum maximal matching problem.
Note that any coloring of a graph with the minimum number of colors must be a complete coloring, so minimizing the number of colors in a complete coloring is just a restatement of the standard graph coloring problem.
Read more about this topic: Complete Coloring
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