The 3-path-connectivity of pancake graphs

Let G be a simple connected graph with vertex set V(G) and edge set E(G). For \(\Phi \subseteq V(G)\) $\mathrm{\Phi }\subseteq V(G)$ , a path that includes all vertices of \(\Phi\) $\mathrm{\Phi }$ is referred to an \(\Phi\) $\mathrm{\Phi }$ -path of G. Two \(\Phi\) $\mathrm{\Phi }$ -paths \(S_{1}\) $S_1$ and \(S_{2}\) $S_2$ of G are internally disjoint if \(V(S_{1})\cap V(S_{2})=\Phi\) $V(S_1)\cap V(S_2)=\mathrm{\Phi }$ and \(E(S_{1})\cap E(S_{2})=\emptyset\) $E(S_1)\cap E(S_2)=\mathrm{\varnothing }$ . Let \(\pi _{G}(\Phi )\) ${\pi }_G(\mathrm{\Phi })$ be the maximum number of internally disjoint \(\Phi\) $\mathrm{\Phi }$ -paths. For an integer k with \(k\ge\) $k\ge$ 2, the k-path-connectivity \(\pi _{k}(G)\) ${\pi }_k(G)$ is defined as the minimum \(\pi _{G}(\Phi )\) ${\pi }_G(\mathrm{\Phi })$ over all k-subsets of V(G). In this paper, we determine 3-path-connectivity of the pancake graphs \(P_{n}\) $P_n$ . By analyzing the structural characteristics of \(P_{n}\) $P_n$ , we show that \(\pi _{3}(P_{n})\) ${\pi }_3(P_n)$ = \(\left\lfloor \frac{3(n-1)-1}{4} \right\rfloor\) $⌊\frac{3(n-1)-1}{4}⌋$ where \(n\ge 3\) $n\ge 3$ .