The 3-path-connectivity of the hypercubes

Abstract

Let G be a connected simple graph with vertex set V ( G ) and edge set E ( G ) . For S ⊆ V ( G ) , let π G ( S ) and κ G ( S ) denote the maximum number of internally disjoint S -paths and S -trees, respectively, in G . For an integer k with k ≥ 2 , the k -path-connectivity π k ( G ) (resp. k -tree-connectivity κ k ( G ) ) is defined as the minimum π G ( S ) (resp. κ G ( S ) ) over all k -subsets S of V ( G ) . It is proved that deciding whether π G ( S ) ≥ k is NP-complete for a given S in Li et al. (2021). In this paper, the upper bound of π 3 ( Q n ) is gotten by using the result π 3 ( G ) ≤ ⌊ 3 k − r 4 ⌋ for a k -regular graph G , where r = max { | N G ( x ) ∩ N G ( y ) ∩ N G ( z ) | : { x , y , z } ⊆ V ( G ) } . Furthermore, we consider the 3-path-connectivity of the n -dimensional hypercube Q n and prove that π 3 ( Q n ) = ⌊ 3 n − 1 4 ⌋ for n ≥ 2 , which implies that the upper bound for Q n is tight.