Internally disjoint trees in the line graph and total graph of the complete bipartite graph

Abstract

Let G be a connected graph, S⊆V(G) and |S|≥2, a tree T in G is called an S-tree if S⊆V(T). Two S-trees T1 and T2 are called internally disjoint if E(T1)∩E(T2)=∅ and V(T1)∩V(T2)=S. For an integer r with 2≤r≤n, the generalizedr-connectivityκr(G) of a graph G is defined as κr(G)=min{κG(S)|S⊆V(G) and |S|=r}, where κG(S) denotes the maximum number k of internally disjoint S-trees in G. In this paper, we consider the generalized 4-connectivity of the line graph L(Km,n) and total graph T(Km,n) of the complete bipartite graph Km,n with 2≤m≤n. The results that κ4(L(Km,n))=m+n−3 for 2≤m≤3 and κ4(L(Km,n))=m+n−4 for m≥4 are obtained by determining κ4(Km×Kn). In addition, we obtain that κ4(T(Km,m))=δ(T(Km,m))−2=2m−2 for m≥2. These results improve the known results about the generalized 3-connectivity of L(Km,n) and T(Km,m) in [Appl. Math. Comput. 347 (2019) 645–652].