On the family of [formula omitted]-regular graphs with Grundy number [formula omitted

Abstract

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 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 the partial Grundy number.