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 kappa(S) denote the maximum number l of edge-disjoint trees T-1, T-2,...,T-l in G such that V(T-i) boolean AND V(T-j) = 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 kappa(k)(G), of G is defined by kappa(k)(G) =min{kappa(S)}, where the minimum is taken over all k-subsets S of V(G). Thus kappa(2)(G) = kappa(G), where kappa(G) is the connectivity of G. Moreover, kappa(n)(G) is the maximum number of edge-disjoint spanning trees of G. This paper mainly focus on the k-connectivity of complete bipartite graphs K-a,K-b, where 1 <= a <= b. First, we obtain the number of edge-disjoint spanning trees of K-a,K-b, which is lfloor ab/a+b-1rfloor, and specifically give the lfloor ab/a+b-1rfloor edge-disjoint spanning trees. Then, based on this result, we get the k-connectivity of K-a,K-b for all 2 <= k <= a + b. Namely, if k > b - a + 2 and a - b + k is odd then kappa(k)(K-a,K-b) = a+b-k+1/2 + lfloor (a-b+k-1)(b-a+k-1)/4(k-1)rfloor, if k > b - a + 2 and a - b + k is even then kappa(k)(K-a,K-b) = a+b-k/2 + lfloor(a-b+k)(b-a+k)/4(k-1)rfloor, and if k <= b - a + 2 then kappa(k)(K-a,K-b) = a.