The generalized connectivity of complete bipartite graphs

Let G be a nontrivial connected graph of order n, and k 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. 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. Moreover, κn(G) is the maximum number of edge-disjoint spanning trees of G. This paper mainly focus on the k-connectivity of complete bipartite graphs Ka,b. First, we obtain the number of edge-disjoint spanning trees ofKa,b, which is ⌊ ab a+b−1⌋, and specifically give the ⌊ ab a+b−1⌋ edge-disjoint spanning trees. Then based on this result, we get the k-connectivity of Ka,b for all 2 ≤ k ≤ a+b. Namely, if k > b−a+2 and a−b+k is odd then κk(Ka,b) = a+b−k+1 2 +⌊ (a−b+k−1)(b−a+k−1) 4(k−1) ⌋, if k > b−a+2 and a− b+ k is even then κk(Ka,b) = a+b−k 2 + ⌊ (a−b+k)(b−a+k) 4(k−1) ⌋, and if k ≤ b− a+2 then κk(Ka,b) = a.