The Toughness of Kneser Graphs

TL;DR

This paper investigates the toughness of Kneser graphs, extending known results to new parameter ranges and conjecturing a general formula for their toughness related to maximum independent sets.

Contribution

It determines the toughness of Kneser graphs for specific parameters and extends previous results, proposing a conjecture for all larger cases.

Findings

Toughness of Kneser graphs $K(n,k)$ for $k=3,4$ and large $n$ is explicitly calculated.

For $k eq 2$, the toughness equals the toughness of the complement of a maximum independent set.

The results support a conjecture that this toughness value holds for all $k eq 2$ and sufficiently large $n$.

Abstract

The \textit{toughness} $t(G)$ of a graph $G$ is a measure of its connectivity that is closely related to Hamiltonicity. Brouwer proved the lower bound $t(G) > \ell / \lambda - 2$ on the toughness of any connected $\ell$-regular graph, where $ \lambda$ is the largest nontrivial eigenvalue of the adjacency matrix. He conjectured that this lower bound can be improved to $\ell / \lambda-1$ and this conjecture is still open. Brouwer also observed that many families of graphs (in particular, those achieving equality in the Hoffman ratio bound for the independence number) have toughness exactly $\ell / \lambda$. Cioab\u{a} and Wong confirmed Brouwer's observation for several families of graphs, including Kneser graphs $K(n,2)$ and their complements, with the exception of the Petersen graph $K(5,2)$. In this paper, we extend these results and determine the toughness of Kneser graphs $K(n,k)$ when $k\in \{3,4\}$ and $n\geq 2k+1$ as well as for $k\geq 5$ and sufficiently large $n$ (in terms of $k$). In all these cases, the toughness is attained by the complement of a maximum independent set and we conjecture that this is the case for any $k\geq 5$ and $n\geq 2k+1$.