  A Grundy k-coloring of a graph G, is a vertex k-coloring of G such that for
each two colors i and j with i < j, every vertex of G colored by j has a
neighbor with color i. The Grundy chromatic number (G), is the largest integer
k for which there exists a Grundy k-coloring for G. In this note we first give
an interpretation of (G) in terms of the total graph of G, when G is the
complement of a bipartite graph. Then we prove that determining the Grundy
number of the complement of bipartite graphs is an NP-Complete problem
