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, . . . , V k and every vertex of Vi is adjacent to at least one vertex in Vj , for every j < i. The objects which are studied in this article are families of r-regular graphs such that Γ(G) = r + 1. Using the notion of independent module, a characterization of this family is given for r = 3. Moreover, we determine classes of graphs in this family, in particular the class of r-regular graphs without induced C4, for r ≤ 4. Furthermore, our propositions imply results on partial Grundy number.