On the generalized (edge-)connectivity of graphs

The generalized k-connectivity κk(G) of a graph G was introduced by Chartrand et al. in 1984. It is natural to introduce the concept of generalized k-edge-connectivity, λk(G). For general k, the generalized k-edgeconnectivity of a complete graph is obtained. For k ≥ 3, tight upper and lower bounds of κk(G) and λk(G) are given for a connected graph G of order n, namely, 1 ≤ κk(G) ≤ n− k2 and 1 ≤ λk(G) ≤ n− k2 . Moreover, graphs of order n such that κk(G) = n− k2 and λk(G) = n− k2 are characterized. Nordhaus-Gaddum-type results for the generalized kconnectivity are also obtained. For k = 3, we study the relation between the edge-connectivity and the generalized 3-edge-connectivity of a graph. Upper and lower bounds of λ3(G) for a graph G in terms of the edgeconnectivity λ of G are obtained, that is, 3λ−2 4 ≤ λ3(G) ≤ λ, and two graph classes are given showing that the upper and lower bounds are tight. From these bounds, we obtain λ(G) − 1 ≤ λ3(G) ≤ λ(G) if G is a connected planar graph, and we also obtain the relation between the generalized 3-connectivity and generalized 3-edge-connectivity of a graph and its line graph.