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

TL;DR

The paper determines the exact 3-path-connectivity of CW_n, the wheel-generated Cayley graph on the symmetric group, by developing a parity-aware structural lemma that connects any three vertices with a large collection of internally disjoint Ω-paths. It combines a decomposition of CW_n into BS_{n-1} copies, vertex/edge-neighborhood analyses, and a constructive path-packing argument to achieve tight bounds. The main result, $π_3(CW_n)=\left\lfloor\frac{6n-9}{4}\right\rfloor$ for all $n\ge4$, advances understanding of path-connectivity in Cayley graphs and informs reliability considerations for such interconnection networks.

Abstract

Let $G = (V(G), E(G))$ be a simple connected graph and $Ω$ a subset of $ V(G)$ with $|Ω|\geq2$. An $Ω$-path in $G$ is a path that connects all vertices of $Ω$. Two $Ω$-paths $P_i$ and $P_j$ are said to be internally disjoint if $V(P_i)\cap V(P_j)=Ω$ and $E(P_i)\cap E(P_j)=\emptyset$. Denote $π_G(Ω)$ by the maximum number of internally disjoint $Ω$-paths in $G$. For an integer $k\geq2$, the $k$-path-connectivity $π_k(G)$ of $G$ is defined as $\min\{π_G(Ω)\midΩ\subseteq V(G)$ and $|Ω|=k\}$. 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 $π_3(CW_n)=\lfloor\frac{6n-9}4\rfloor$ for all $n\geq4$.

Figures (9)

• Figure 1: Illustrations of Lemma 2.6 Case 2
• Figure 2: Illustrations of Theorem 3.1
• Figure 3: Illustration of Subcase 1.2.1
• Figure 4: Illustration of Subcase 1.2.2
• Figure 5: Illustration of Case 2

Theorems & Definitions (11)

• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 2.4
• Lemma 2.5
• Lemma 2.6
• Lemma 2.7
• Lemma 2.8
• Lemma 2.9
• Theorem 3.1