The generalized 3-connectivity of Lexicographic product graphs

HAL is a multidisciplinary open access archive for the deposit and dissemination of scientific research documents, whether they are published or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers. L'archive ouverte pluridisciplinaire HAL, est destinée au dépôt età la diffusion de documents scientifiques de niveau recherche, publiés ou non, ´ emanant desétablissements d'enseignement et de recherche français oú etrangers, des laboratoires publics ou privés. The generalized k-connectivity κ k (G) of a graph G, first introduced by Hager, is a natural generalization of the concept of (vertex-)connectivity. Denote by G • H and G2H the lexicographic product and Cartesian product of two graphs G and H, respectively. In this paper, we prove that for any two connected graphs G and H, κ3(G • H) ≥ κ3(G)|V (H)|. We also give upper bounds for κ3(G2H) and κ3(G • H). Moreover, all the bounds are sharp.