  A coloring of a graph G = (V,E) is a partition {V1, V2, . . ., Vk} of V into
independent sets or color classes. A vertex v Vi is a Grundy vertex if it is
adjacent to at least one vertex in each color class Vj . A coloring is a Grundy
coloring if every color class contains at least one Grundy vertex, and the
Grundy number of a graph is the maximum number of colors in a Grundy coloring.
We derive a natural upper bound on this parameter and show that graphs with
sufficiently large girth achieve equality in the bound. In particular, this
gives a linear time algorithm to determine the Grundy number of a tree.
