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

Abstract

Let $G=(V(G),E(G))$ be a simple connected graph and $\hat{I}\text{c◯}$ a subset of $V(G)$ with $|\hat{I}\text{c◯}|\ge 2$. An $\hat{I}\text{c◯}$-path in $G$ is a path that connects all vertices of $\hat{I}\text{c◯}$. Two $\hat{I}\text{c◯}$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=\hat{I}\text{c◯}$ and $E(P_i)\cap E(P_j)=\varnothing$. Denote $\ddot{I}{€}_G(\hat{I}\text{c◯})$ by the maximum number of internally disjoint $\hat{I}\text{c◯}$-paths in $G$. For an integer $k\ge 2$, the $k$-path-connectivity $\ddot{I}{€}_k(G)$ of $G$ is defined as $\min \{\ddot{I}{€}_G(\hat{I}\text{c◯})\mid \hat{I}\text{c◯}\subseteq V(G)$ and $|\hat{I}\text{c◯}|=k\}$. 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 $\ddot{I}{€}_3(C{W}_n)=\lfloor \frac{6n-9}{4}\rfloor$ for all $n\ge 4$.