The 3-path-connectivity of the augmented cubes

Connectivity is a key concept in graph theory, used to evaluate network robustness.To better reflect fault tolerance in complex systems, Hager introduced the k-path-connectivity, π k (G), as an extension of classical connectivity.Given a connected simple graph G = (V, E) and a subset D ⊆ V with |D| ≥ 2, a D-path is a path that includes all vertices in D. A collection of such paths is internally disjoint if they intersect only at the vertices of D and share no edges.The maximum number of internally disjoint D-paths in G is denoted π G (D), and the k-path-connectivity is defined asIn this paper, we investigate the 3-path-connectivity of the augmented cube AQ n , a variant of the hypercube known for its enhanced symmetry and fault-tolerant structure.We establish the exact value of π 3 (AQ n ) and show thatif n is even,