Kneser Graph

In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1956.

Examples

The Kneser graph is the complete graph on vertices.

The Kneser graph is the complement of the line graph of the complete graph on vertices.

The Kneser graph is the odd graph ; in particular is the Petersen graph (see top right figure).

The Kneser graph , visualized on the right.

Properties

Basic properties

The Kneser graph $K(n,k)$ has $\tbinom$ vertices. Each vertex has exactly $\tbinom$ neighbors.

The Kneser graph is vertex transitive and arc transitive. When $k=2$, the Kneser graph is a strongly regular graph, with parameters $( \tbinom, \tbinom, \tbinom, \tbinom )$. However, it is not strongly regular when $k>2$, as different pairs of nonadjacent vertices have different numbers of common neighbors depending on the size of the intersection of the corresponding pairs of sets.

Because Kneser graphs are regular and edge-transitive, their vertex connectivity equals their degree, except for $K(2k,k)$ which is disconnected. More precisely, the connectivity of $K(n,k)$ is $\tbinom,$ the same as the number of neighbors per vertex.

Chromatic number

As conjectured, the chromatic number of the Kneser graph $K(n,k)$ for $n\geq 2k$ is exactly ; for instance, the Petersen graph requires three colors in any proper coloring. This conjecture was proved in several ways.

*László Lovász proved this in 1978 using topological methods, giving rise to the field of topological combinatorics.
*Soon thereafter Imre Bárány gave a simple proof, using the Borsuk–Ulam theorem and a lemma of David Gale.
*Joshua E. Greene won the 2002 the Morgan Prize for outstanding undergraduate research for his further simplified but still topological proof.
*In 2004, Jiří Matoušek found a purely combinatorial proof.
In contrast, the fractional chromatic number of these graphs is $n/k$.
When $n< 2k$, $K(n,k)$ has no edges and its chromatic number is 1.

Hamiltonian cycle

The Kneser graph contains a Hamiltonian cycle if
$n\geq \frac \left (3k+1+\sqrt \right ).$ Since
$\frac \left (3k+1+\sqrt \right )< \left (\frac \right) k+1$
holds for all $k$ this condition is satisfied if
$n\geq \left (\frac \right) k+1 \approx 2.62k+1.$

The Kneser graph contains a Hamiltonian cycle if there exists a non-negative integer ''a'' such that $n=2k+2^a$. In particular, the odd graph has a Hamiltonian cycle if . With the exception of the Petersen graph, all connected Kneser graphs with are Hamiltonian.

Cliques

When , the Kneser graph contains no triangles. More generally, when it does not contain cliques of size , whereas it does contain such cliques when . Moreover, although the Kneser graph always contains cycles of length four whenever , for values of close to the shortest odd cycle may have variable length.

Diameter

The diameter of a connected Kneser graph is
$\left\lceil \frac \right\rceil + 1.$

Spectrum

The spectrum of the Kneser graph consists of ''k'' + 1 distinct eigenvalues:
$\lambda_j=(-1)^j\binom, \qquad j=0, \ldots,k.$
Moreover $\lambda_j$ occurs with multiplicity $\tbinom-\tbinom$ for $j >0$ and $\lambda_0$ has multiplicity 1.

Independence number

The Erdős–Ko–Rado theorem states that the independence number of the Kneser graph for $n\geq 2k$ is
$\alpha(K(n,k))=\binom.$

Related graphs

The Johnson graph is the graph whose vertices are the -element subsets of an -element set, two vertices being adjacent when they meet in a -element set. The Johnson graph is the complement of the Kneser graph . Johnson graphs are closely related to the Johnson scheme, both of which are named after Selmer M. Johnson.

The generalized Kneser graph has the same vertex set as the Kneser graph , but connects two vertices whenever they correspond to sets that intersect in or fewer items. Thus .

The bipartite Kneser graph has as vertices the sets of and items drawn from a collection of elements. Two vertices are connected by an edge whenever one set is a subset of the other. Like the Kneser graph it is vertex transitive with degree $\tbinom.$ The bipartite Kneser graph can be formed as a bipartite double cover of in which one makes two copies of each vertex and replaces each edge by a pair of edges connecting corresponding pairs of vertices. The bipartite Kneser graph is the Desargues graph and the bipartite Kneser graph is a crown graph.

References

Notes

Works cited

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External links

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{{DEFAULTSORT:Kneser Graph
Parametric families of graphs
Regular graphs