Well-colored graph
From Wikipedia, the free encyclopedia
In graph theory, a subfield of mathematics, a well-colored graph is an undirected graph for which greedy coloring uses the same number of colors regardless of the order in which colors are chosen for its vertices. That is, for these graphs, the chromatic number (minimum number of colors) and Grundy number (maximum number of greedily-chosen colors) are equal.[1]
The well-colored graphs include the complete graphs and odd-length cycle graphs (the graphs that form the exceptional cases to Brooks' theorem) as well as the complete bipartite graphs and complete multipartite graphs.
The simplest example of a graph that is not well-colored is a four-vertex path. Coloring the vertices in path order uses two colors, the optimum for this graph. However, coloring the ends of the path first (using the same color for each end) causes the greedy coloring algorithm to use three colors for this graph. Because there exists a non-optimal vertex ordering, the path is not well-colored.[2][3]
