A complete coloring is a coloring of the vertices of a simple graph so that for every pair of colors there are a pair of vertices with those colors that are adjacent.
The achromatic number of a graph is the greatest number of colors which can be used in a complete coloring.
Does there exist a systematic approach for calculating the achromatic number of a graph? I could express the conditions as a complete $k$-coloring as a boolean SAT problem, but I was wondering if there was a better way. Maybe even software for this purpose, already written?