The super-connectivity of Kneser graph KG(n,3)

TL;DR

This paper proves the conjecture that the super-connectivity of Kneser graphs KG(n,3) equals their minimum degree, confirming a specific case of a broader conjecture in graph theory.

Contribution

The paper establishes the super-connectivity of KG(n,3), confirming a conjecture for this specific case and advancing understanding of the connectivity properties of Kneser graphs.

Findings

Super-connectivity of KG(n,3) equals its minimum degree.

Confirmed the conjecture by Boruzanli and Gauci for k=3.

Enhanced understanding of the connectivity structure of Kneser graphs.

Abstract

A vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected. A super vertex cut $S$ of a connected graph $G$ is a subset of vertices of $G$ whose deletion makes $G$ disconnected and there is no isolated vertex in each component of $G-S$. The super-connectivity of graph $G$ is the size of the minimum super vertex cut of $G$. Let $KG(n,k)$ be the Kneser graph whose vertices set are the $k$-subsets of $\{1,\cdots,n\}$, where $k$ is the number of labels of each vertex in $G$. We aim to show that the conjecture from Boruzanli and Gauci \cite{EG19} on the super-connectivity of Kneser graph $KG(n,k)$ is true when $k=3$.