Two kinds of generalized connectivity of dual cubes

Let S⊆V(G) and κG(S) denote the maximum number k of edge-disjoint trees T1,T2,…,Tk in G such that V(Ti)⋂V(Tj)=S for any i,j∈{1,2,…,k} and i≠j. For an integer r with 2≤r≤n, the generalizedr-connectivity of a graph G is defined as κr(G)=min{κG(S)|S⊆V(G) and |S|=r}. The r-component connectivity cκr(G) of a non-complete graph G is the minimum number of vertices whose deletion results in a graph with at least r components. These two parameters are both generalizations of traditional connectivity. Except hypercubes and complete bipartite graphs, almost all known κr(G) are about r=3. In this paper, we focus on κ4(Dn) of dual cube Dn. We first show that κ4(Dn)=n−1 for n≥4. As a corollary, we obtain that κ3(Dn)=n−1 for n≥4. Furthermore, we show that cκr+1(Dn)=rn−r(r+1)2+1 for n≥2 and 1≤r≤n−1.