k-Coloring

A -coloring of a graph  is a vertex coloring that is an assignment of one of  possible colors to each vertex of  (i.e., a vertex coloring) such that no two adjacent vertices receive the same color.

Note that a -coloring may contain fewer than  colors for .

A -coloring of a graph can be computed using MinimumVertexColoring[g, k] in the Wolfram Language package Combinatorica` , and all -colorings may be computed using MinimumVertexColoring[g, k, All] (where, however, the command returns colorings that differ by permutation of colors only a single time).

The number of distinct -colorings (where permutations of colors are counted separately) of a graph  is given by , where  is the chromatic polynomial of .

REFERENCES

Saaty, T. L. and Kainen, P. C. The Four-Color Problem: Assaults and Conquest. New York: Dover, p. 13, 1986.