The Grundy Number of a graph G, denoted by Γ(G), is the largest k such that there exists a partition of V (G), into k independent sets V1, ..., Vk and every vertex in Vi is adjacent to at least one vertex in Vj , for every j < i. We consider the following question for which we give some partial answers: Does the equality Γ(G) = r + 1 hold for any r-regular graph without induced 4-cycle?