The generalized connectivity of complete equipartition 3-partite 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 ` of edge-disjoint trees T1, T2, . . . , T` in G such that V (Ti) ∩ V (Tj) = S for every pair of distinct integers i, j with 1 ≤ i, j ≤ `. Chartrand et al. generalized the concept of connectivity as follows: The kconnectivity of G, denoted by κk(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; whereas, κn(G) is the maximum number of edge-disjoint spanning trees contained in G. This paper mainly focuses on the k-connectivity of complete equipartition 3-partite graphs K b , where b ≥ 2 is an integer. First, we obtain the number of edge-disjoint spanning trees of a general complete 3-partite graph Kx,y,z, which is b x+y+z−1 c. Then, based on this result, we get the k-connectivity of K b for all 3 ≤ k ≤ 3b. Namely, κk(K b ) =    b d k 2 3 e+k2−2kb 2(k−1) c+ 3b− k if k ≥ 3b 2 ; b 3b 2 c if k < 3b 2 and k = 0 (mod 3); b 3bk+3b−k+1 2k+1 c if 3b 4 < k < 3b 2 and k = 1 (mod 3); b 3bk+6b−2k+1 2k+2 c if b ≤ k < 3b 2 and k = 2 (mod 3); b 3b+1 2 c otherwise.