A linear algorithm for the grundy number of a tree

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 for every j <i. 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.