The 3-path-connectivity of the augmented cubes

Connectivity is a cornerstone concept in graph theory, essential for evaluating the robustness of networks against failures. To better capture fault tolerance in complex systems, researchers have extended classical connectivity notions, one such extension being the $k$-path-connectivity, $\pi_k(G)$, introduced by Hager. Given a connected simple graph $G = (V, E)$ and a subset $D \subseteq V$ with $|D| \geq 2$, 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 $\pi_G(D)$, and the $k$-path-connectivity is defined as $\pi_k(G) = \min \{ \pi_G(D) \mid D \subseteq V(G),\ |D| = k\}$. In 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 $\pi_3(AQ_n)$ and show that: $$\pi_3(AQ_n) = \begin{cases} \frac{3n}{2} - 2, & \text{if } n \text{ is even}, \frac{3(n - 1)}{2} - 1, & \text{if } n \text{ is odd}. \end{cases}$$