Kneser graph

Definition

Suppose $n$ and $k$ are positive integers. The Kneser graph $K{G}_{n,k}$ is an undirected graph defined as follows. Fix a set of size $n$ (we usually take the set as $\{1,2,\dots ,n\}$ for convenience). Then:

1. The vertex set of $K{G}_{n,k}$ is the collection of $k$-element subsets of the fixed set of size $n$.
2. The edge set of $K{G}_{n,k}$ is defined as follows: two vertices of $K{G}_{n,k}$ are adjacent if and only if they are disjoint when viewed as subsets of the $n$-element set.

Particular cases

Classes of cases

Condition on $n$ and $k$	Conclusion
$k>n$	The vertex set is empty, so the graph is a graph on no vertices
$k=n$	one-point graph
$n/2<k<n$	The vertex set is non-empty, but the edge set is empty, so the graph is an empty graph
$k=n/2$	In this case, the graph is a matching graph: it is a disjoint union of 2-cliques, with one 2-clique for each partition of the $n$-set into two disjoint pieces of equal size.
$1<k<n/2$	This is the interesting case. In this case, the graph is connected and non-empty, but is not a complete graph.
$k=1$	complete graph $K_n$
$k=0$	one-point graph

First few nontrivial cases

As indicated above, the interesting cases are where $1<k<n/2$. We list the first few of these:

$n$	$k$	Kneser graph $K{G}_{n,k}$
5	2	Petersen graph
6	2	Fill this in later