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

TL;DR

This paper determines the exact 3-path-connectivity of Cayley graphs generated by wheel graphs, showing it equals the floor of (6n-9)/4 for all n ≥ 4, thus advancing understanding of graph connectivity properties.

Contribution

The paper provides a precise formula for the 3-path-connectivity of Cayley wheel graphs, a novel result in graph theory.

Findings

Exact formula for $rac{6n-9}{4}$ for all n ≥ 4

First determination of 3-path-connectivity for these Cayley graphs

Enhances understanding of connectivity in wheel-generated Cayley graphs

Abstract

Let $G = (V(G), E(G))$ be a simple connected graph and $\Omega$ a subset of $ V(G)$ with $|\Omega|\geq2$. An $\Omega$-path in $G$ is a path that connects all vertices of $\Omega$. Two $\Omega$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=\Omega$ and $E(P_i)\cap E(P_j)=\emptyset$. Denote $\pi_G(\Omega)$ by the maximum number of internally disjoint $\Omega$-paths in $G$. For an integer $k\geq2$, the $k$-path-connectivity $\pi_k(G)$ of $G$ is defined as $\min\{\pi_G(\Omega)\mid\Omega\subseteq V(G)$ and $|\Omega|=k\}$. Let $CW_n$ denote the Cayley graph generated by the $n$-vertex wheel graph. In this paper, we investigate the $3$-path-connectivity of $CW_n$ and prove that $\pi_3(CW_n)=\lfloor\frac{6n-9}4\rfloor$ for all $n\geq4$.