On the dissociation number of Kneser graphs
| Autor: | Brešar, Boštjan, Dravec, Tanja |
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| Rok vydání: | 2021 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | 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 $diss(K_{2k+1,k})={2k \choose k}$. Comment: 9 pages, 1 figure |
| Databáze: | arXiv |
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