Brelaz's Heuristic Algorithm

An algorithm which can be used to find a good, but not necessarily minimal, edge or vertex coloring for a graph. However, the algorithm does minimally color complete k-partite graphs.

Brelaz's algorithm can be applied using BrelazColoring[g] in the Wolfram Language package Combinatorica` , and a guaranteed minimal vertex coloring can be found for small graphs using backtracking with MinimumVertexColoring[g].

REFERENCES

Brelaz, D. "New Methods to Color the Vertices of a Graph." Comm. ACM 22, 251-256, 1979.

Skiena, S. "Finding a Vertex Coloring." §5.5.3 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 214-215, 1990.