The 3-path-connectivity of the k-ary n-cube

Let G be a connected simple graph with vertex set V(G). Let Omega be a subset with cardinality at least two of V (G ) . A path containing all vertices of Omega is said to be an Omega-path of G . Two Omega-paths T-1 and T-2 of G are internally disjoint if V(T-1) boolean AND V(T-2) = Omega and E(T-1) boolean AND E(T-2) = theta. For an integer l with 2 <= l, the l-path-connectivity pi(l)(G) is defined as pi(l)(G) =min{ pi(G)(Omega) | Omega subset of V (G ) and | Omega| = l}, where pi G (Omega) represents the maximum number of internally disjoint Omega-paths. In this paper, we completely determine 3-path -connectivity of the k-ary n-cube Q(n)(k). By deeply exploring the structural proprieties of Q(k)(n) , we show that pi(3) (Q(k)(n)) = [6 n-1/4] with n >= 1 and k >= 3. (c) 2022 Elsevier Inc. All rights reserved.