On the dissociation number of Kneser graphs

TL;DR

This paper investigates the dissociation number of Kneser graphs, providing exact values for specific cases, establishing bounds, and exploring the relationship between dissociation and independence numbers.

Contribution

It offers new exact formulas and bounds for the dissociation number of Kneser graphs, including specific cases like $K_{n,2}$ and odd graphs, and improves existing upper bounds.

Findings

Dissociation number of $K_{n,2}$ equals $ ext{max} -1,6$.

For large $n$, dissociation number equals independence number.

Exact dissociation number for odd Kneser graphs $K_{2k+1,k}$ is ${2k race k}$.

Abstract

A set $D$ of vertices of a graph $G$ is a dissociation set if each vertex of $D$ has at most one neighbor in $D$. The dissociation number of $G$, $diss(G)$, is the cardinality of a maximum dissociation set in a graph $G$. In this paper we study dissociation in the well-known class of Kneser graphs $K_{n,k}$. In particular, we establish that the dissociation number of Kneser graphs $K_{n,2}$ equals $\max{\{n-1,6\}}$. We show that for any $k \geq 2$, there exists $n_0 \in \mathbb{N}$ such that $diss(K_{n,k})=\alpha(K_{n,k})$ for any $n \geq n_0$. We consider the case $k=3$ in more details and prove that $n_0=8$ in this case. Then we improve a trivial upper bound $2\alpha(K_{n,k})$ for the dissociation number of Kneser graphs $K_{n,k}$ by using Katona's cyclic arrangement of integers from $\{1,\ldots , n\}$. Finally we investigate the odd graphs, that is, the Kneser graphs with $n=2k+1$. We prove that $diss(K_{2k+1,k})={2k \choose k}$.