Ju n 20 09 Sharp bounds for the generalized connectivity κ 3 ( G ) ∗

Let G be a nontrivial connected graph of order n and let k be an integer with 2 ≤ k ≤ n. For a set S of k vertices of G, let κ(S) denote the maximum number l of edge-disjoint trees T1, T2, . . . , Tl in G such that V (Ti)∩V (Tj) = S for every pair i, j of distinct integers with 1 ≤ i, j ≤ l. A collection {T1, T2, . . . , Tl} of trees in G with this property is called an internally disjoint set of trees connecting S. Chartrand et al. generalized the concept of connectivity as follows: The k-connectivity, denoted by κk(G), of G is defined by κk(G) =min{κ(S)}, where the minimum is taken over all k-subsets S of V (G). Thus κ2(G) = κ(G), where κ(G) is the connectivity of G. In general, the investigation of κk(G) is very difficult. We therefore focus on the investigation on κ3(G) in this paper. We study the relation between the connectivity and the 3-connectivity of a graph. First we give sharp upper and lower bounds of κ3(G) for general graphs G, and construct two kinds of graphs which attain the upper and lower bound, respectively. We then show that if G is a connected planar graph, then κ(G)−1 ≤ κ3(G) ≤ κ(G), and give some classes of graphs which attain the bounds. In the end we show that the problem whether κ(G) = κ3(G) for a planar graph G can be solved in polynomial time.