Well-colored graph


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 and Grundy number are equal.

Examples

The well-colored graphs include the complete graphs and odd-length cycle graphs 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 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.

Complexity

A graph is well-colored if and only if does not have two vertex orderings for which the greedy coloring algorithm produces different numbers of colors. Therefore, recognizing non-well-colored graphs can be performed within the complexity class NP. On the other hand, a graph has Grundy number or more if and only if the
graph obtained from by adding a -vertex clique is well-colored. Therefore, by a reduction from the Grundy number problem,
it is NP-complete to test whether these two orderings exist.
It follows that it is co-NP-complete to test whether a given graph is well-colored.

Related properties

A graph is hereditarily well-colored if every induced subgraph is well-colored. The hereditarily well-colored graphs are exactly the cographs, the graphs that do not have a four-vertex path as an induced subgraph.