Grundy number

In graph theory, the Grundy number or Grundy chromatic number of an undirected graph is the maximum number of colors that can be used by a greedy coloring strategy that considers the vertices of the graph in sequence and assigns each vertex its first available color, using a vertex ordering chosen to use as many colors as possible. Grundy numbers are named after P. M. Grundy, who studied an analogous concept for directed graphs in 1939. The undirected version was introduced by Christen & Selkow (1979).

For example, for a path graph with four vertices, the chromatic number is two but the Grundy number is three: if the two endpoints of the path are colored first, the greedy coloring algorithm will use three colors for the whole graph.

Zaker (2006) defines a sequence of graphs called t-atoms, with the property that a graph has Grundy number at least t if and only if it contains a t-atom. Each t-atom is formed from an independent set and a (t − 1)-atom, by adding one edge from each vertex of the (t − 1)-atom to a vertex of the independent set, in such a way that each member of the independent set has at least one edge incident to it. A Grundy coloring of a t-atom can be obtained by coloring the independent set first with the smallest-numbered color, and then coloring the remaining (t − 1)-atom with an additional t − 1 colors. For instance, the only 1-atom is a single vertex, and the only 2-atom is a single edge, but there are two possible 3-atoms: a triangle and a four-vertex path.

For a graph with n vertices and degeneracy d, the Grundy number is O(d log n). In particular, for graphs of bounded degeneracy (such as planar graphs) or graphs for which the chromatic number and degeneracy are bounded within constant factors of each other (such as chordal graphs) the Grundy number and chromatic number are within a logarithmic factor of each other. For interval graphs, the chromatic number and Grundy number are within a factor of 8 of each other.

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