3-path-connectivity of Cayley graphs generated by wheel graphs

Let $G=(V(G),E(G))$ be a simple connected graph and $\mathrm{\Omega }$ a subset of $V(G)$ with $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }\ge 2$. An $\mathrm{\Omega }$-path in $G$ is a path that connects all vertices of $\mathrm{\Omega }$. Two $\mathrm{\Omega }$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=\mathrm{\Omega }$ and $E(P_i)\cap E(P_j)=\mathrm{\varnothing }$. Denote ${\pi }_G(\mathrm{\Omega })$ by the maximum number of internally disjoint $\mathrm{\Omega }$-paths in $G$. For an integer $k\ge 2$, the $k$-path-connectivity ${\pi }_k(G)$ of $G$ is defined as $\min \{{\pi }_G(\mathrm{\Omega })\mid \mathrm{\Omega }\subseteq V(G)$ and $\mathrm{\mid }\mathrm{\Omega }\mathrm{\mid }=k\}$. Let $C{W}_n$ denote the Cayley graph generated by the $n$-vertex wheel graph. In this paper, we investigate the $3$-path-connectivity of $C{W}_n$ and prove that ${\pi }_3(C{W}_n)=\lfloor \frac{6n-9}{4}\rfloor$ for all $n\ge 4$.