Path connectivity of line graphs and total graphs of complete bipartite graphs

Abstract

Let θ be a subset of vertex set V(G) of a simply connected graph G, two θ-trees P1 and P2 of G are said to be internally disjoint if V(P1)∩V(P2)=θ and E(P1)∩E(P2)=∅. For an integer k≥2, the k-path connectivity πk(G) (resp. k-tree connectivity κk(G)) of a graph G is defined as min{πG(θ)|θ⊆V(G)and|θ|=k} (resp. min{κG(θ)|θ⊆V(G)and|θ|=k}), where πG(θ) (resp. κG(θ)) is the maximum number of pairwise internally disjoint θ-paths (resp. θ-trees) in G. The 3-tree connectivity of the line graph L(Km,n) and total graph T(Km,n) of the complete bipartite graph Km,n are gotten in [Appl. Math. Comput. 347(2019) 645-652]. In this paper, these results are improved from trees to paths. The exact values of the 3-path connectivity for L(Km,n) and T(Km,n) are gotten. That is, π3(L(Km,n))=⌊3m+2n−34⌋−1 for m=3 and odd n, otherwise, π3(L(Km,n))=⌊3m+2n−34⌋ unless m=1 and n=1,2; π3(T(Km,n))=m+⌊m4⌋ for n≥m≥1. In addition, the compact upper bound of π3(G) for a general graph G are gotten.