6533b860fe1ef96bd12c2e29

RESEARCH PRODUCT

On the family ofr-regular graphs with Grundy numberr+1

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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 V 1 , … , V k and every vertex of V i is adjacent to at least one vertex in V j , 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 C 4 , for r ≤ 4 . Furthermore, our propositions imply results on the partial Grundy number.