The 3-path-connectivity of the hypercubes

Let G be a connected simple graph with vertex set V(G) and edge set E(G). For S subset of V(G), let pi(G)(S) and kappa(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 pi(k)(G) (resp. k-tree-connectivity kappa(k)(G)) is defined as the minimum pi(G)(S) (resp. kappa(G)(S)) over all k-subsets S of V(G). It is proved that deciding whether pi(G)(S) >= k is NP-complete for a given S in Li et al. (2021). In this paper, the upper bound of pi(3)(Q(n)) is gotten by using the result pi(3)(G) <= left perpendicular3k-r/4right perpendicular. for a k-regular graph G, where r =max{vertical bar N-G(x) boolean AND N-G(y) boolean AND N-G(z)vertical bar : {x, y, z} subset of V(G)}. Furthermore, we consider the 3-path-connectivity of the n-dimensional hypercube Qn and prove that pi(3)(Q(n)) = left perpendicular3n-1/4right perpendicular for n >= 2, which implies that the upper bound for Q(n) is tight. (C) 2022 Elsevier B.V. All rights reserved.