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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 lineartime algorithm to determine the Grundy number of a tree.
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