Kneser graph

For the Kneser neighborhood graph of unimodular lattices, see Niemeier lattice.

Kneser graph
The Kneser graph $KG 5,2$ , isomorphic to the Petersen graph
Named after	Martin Kneser
Vertices	$\textstyle\binom{n}{k}$
Edges	$\textstyle\binom{n}{k} \binom{n - k}{k} / 2$
Chromatic number	$\left\{\begin{array}{ll}n - 2 k %2B 2 & n \ge 2 k\\ 1 & \text{otherwise}\end{array}\right.$
Properties	$\textstyle\binom{n - k}{k}$ -regular arc-transitive
Notation	$KG n, k$ , $K (n, k)$

In graph theory, the Kneser graph $KG n, k$ is the graph whose vertices correspond to the $k$ -element subsets of a set of $n$ elements, and where two vertices are connected if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1955.

1 Examples
2 Properties
3 Related graphs
4 References
5 External links

Examples

The complete graph on $n$ vertices is the Kneser graph $KG n,1$ .

The Kneser graph $KG 2 n - 1, n - 1$ is known as the odd graph $O n$ ; the odd graph $O 3 = KG 5,2$ is isomorphic to the Petersen Graph.

Properties

The Kneser graph is vertex transitive and edge transitive. Each vertex has exactly $\textstyle\binom{n-k}{k}$ neighbors. However, the Kneser graph is not, in general, a strongly regular graph, as different pairs of nonadjacent vertices have different numbers of common neighbors depending on the size of the intersection of the corresponding pair of sets.
As Kneser (1955) conjectured, the chromatic number of the Kneser graph $KG n, k$ is exactly $n - 2 k + 2$ ; for instance, the Petersen graph requires three colors in any proper coloring. László Lovász (1978) proved this using topological methods, giving rise to the field of topological combinatorics. Soon thereafter Imre Bárány (1978) gave a simple proof, using the Borsuk–Ulam theorem and a lemma of David Gale, and Greene (2002) won the Morgan Prize for a further simplified but still topological proof. Matoušek (2004) found a purely combinatorial proof.
When $n \geq 3 k$ , the Kneser graph $KG n, k$ always contains a Hamiltonian cycle (Chen 2000). Computational searches have found that all connected Kneser graphs for $n \leq 27$ , except for the Petersen graph, are Hamiltonian (Shields 2004).
When $n < 3 k$ , the Kneser graph contains no triangles. More generally, although the Kneser graph always contains cycles of length four whenever $n \geq 2 k + 2$ , for values of $n$ close to $2 k$ the shortest odd cycle may have nonconstant length (Denley 1997).
The diameter of a connected Kneser graph $KG n, k$ is

$\left\lceil \frac{k-1}{n-2k} \right\rceil %2B 1$ (Valencia-Pabon & Vera 2005).

Related graphs

The Johnson graph $J n, k$ is the graph whose vertices are the $k$ -element subsets of an $n$ -element set, two vertices being adjacent when they meet in a $(k - 1)$ -element set. For $k = 2$ the Johnson graph is the complement of the Kneser graph $KG n,2$ . Johnson graphs are closely related to the Johnson scheme, both of which are named after Selmer M. Johnson.

The generalized Kneser graph $KG n, k, s$ has the same vertex set as the Kneser graph $KG n, k$ , but connects two vertices whenever they correspond to sets that intersect in $s$ or fewer items (Denley 1997). Thus $KG n, k,0 = KG n, k$ .

The bipartite Kneser graph $H n, k$ has as vertices the sets of $k$ and $n - k$ items drawn from a collection of $n$ 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 $\textstyle\binom{n-k}{k}$ . The bipartite Kneser graph can be formed as a bipartite double cover of $KG n, k$ in which one makes two copies of each vertex and replaces each edge by a pair of edges connecting corresponding pairs of vertices (Simpson 1991). The bipartite Kneser graph $H 5,2$ is the Desargues graph and the bipartite Kneser graph $H n,1$ is a crown graph.

References

Bárány, Imre (1978), "A short proof of Kneser's conjecture", Journal of Combinatorial Theory, Series A 25 (3): 325–326, doi:10.1016/0097-3165(78)90023-7, MR 0514626 .
Chen, Ya-Chen (2000), "Kneser graphs are Hamiltonian for $n \geq 3 k$ ", Journal of Combinatorial Theory, Series B 80 (1): 69–79, doi:10.1006/jctb.2000.1969, MR 1778200, http://math.la.asu.edu/~cchen/kneser.ps .
Denley, Tristan (1997), "The odd girth of the generalised Kneser graph", European Journal of Combinatorics 18 (6): 607–611, doi:10.1006/eujc.1996.0122, MR 1468332 .
Greene, Joshua E. (2002), "A new short proof of Kneser's conjecture", American Mathematical Monthly 109 (10): 918–920, doi:10.2307/3072460, MR 1941810 .
Kneser, Martin (1955), "Aufgabe 360", Jahresbericht der Deutschen Mathematiker-Vereinigung, 2. Abteilung 58: 27 .
Lovász, László (1978), "Kneser's conjecture, chromatic number, and homotopy", Journal of Combinatorial Theory, Series A 25 (3): 319–324, doi:10.1016/0097-3165(78)90022-5, MR 0514625 .
Matoušek, Jiří (2004), "A combinatorial proof of Kneser's conjecture", Combinatorica 24 (1): 163–170, doi:10.1007/s00493-004-0011-1, MR 2057690 .
Shields, Ian Beaumont (2004), Hamilton Cycle Heuristics in Hard Graphs, Ph.D. thesis, North Carolina State University, http://www.lib.ncsu.edu/theses/available/etd-03142004-013420/ .
Simpson, J. E. (1991), "Hamiltonian bipartite graphs", Proceedings of the Twenty-second Southeastern Conference on Combinatorics, Graph Theory, and Computing (Baton Rouge, LA, 1991), Congressus Numerantium, 85, pp. 97–110, MR 1152123 .
Valencia-Pabon, Mario; Vera, Juan-Carlos (2005), "On the diameter of Kneser graphs", Discrete Mathematics 305 (1–3): 383–385, doi:10.1016/j.disc.2005.10.001, MR 2186709 .

External links

Weisstein, Eric W., "Kneser Graph" from MathWorld.
Weisstein, Eric W., "Odd Graph" from MathWorld.