The super spanning connectivity of arrangement graph

TL;DR

This paper proves that arrangement graphs with certain parameters are super spanning connected, meaning they contain spanning subgraphs with multiple disjoint paths between any two vertices, which enhances their robustness and connectivity properties.

Contribution

The paper establishes that arrangement graphs $A_{n,k}$ are super spanning connected for $n \\ge 4$ and $n-k \\ge 2$, a new result in graph connectivity theory.

Findings

Arrangement graphs are super spanning connected under specified conditions.

The result applies for all $n \\ge 4$ with $n-k \\ge 2$.

Enhances understanding of the robustness of arrangement graphs.

Abstract

A $k$-container $C(u, v)$ of a graph $G$ is a set of $k$ internally disjoint paths between $u$ and $v$. A $k$-container $C(u, v)$ of $G$ is a $k^*$-container if it is a spanning subgraph of $G$. A graph $G$ is $k^*$-connected if there exists a $k^*$-container between any two different vertices of G. A $k$-regular graph $G$ is super spanning connected if $G$ is $i^*$-container for all $1\le i\le k$. In this paper, we prove that the arrangement graph $A_{n, k}$ is super spanning connected if $n\ge 4$ and $n-k\ge 2$.