Kneser graph

Kneser graph
The Kneser graph KG5,2, 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	$\left\{\begin{array}{ll}n - 2 k + 2 & n \ge 2 k\\ 1 & \text{otherwise}\end{array}\right.$
Properties	$\textstyle\binom{n - k}{k}$-regular arc-transitive
Notation	KGn,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 adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1955.

Examples

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

The complement of the line graph of the complete graph on n vertices is the Kneser graph KG_n,2.

The Kneser graph KG_{2n − 1,n − 1} is known as the odd graph O_n; the odd graph O₃ = 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 − 2k + 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 Joshua E. Greene (2002) won the Morgan Prize for a further simplified but still topological proof. Jiří Matoušek (2004) found a purely combinatorial proof.
• When n ≥ 3k, the Kneser graph KG_n,k always contains a Hamiltonian cycle (Chen 2000). Computational searches have found that all connected Kneser graphs for n ≤ 27, except for the Petersen graph, are Hamiltonian (Shields 2004).
• When n < 3k, the Kneser graph contains no triangles. More generally, although the Kneser graph always contains cycles of length four whenever n ≥ 2k + 2, for values of n close to 2k the shortest odd cycle may have nonconstant length (Denley 1997).
• The diameter of a connected Kneser graph KG_n,k is

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

• The graph spectrum of the Kneser graph KG_n,k is given as follows:

For $j=0,...,k$, the eigenvalue $\lambda_j=(-1)^j\binom{n-k-j}{k-j}$ occurs with multiplicity $\binom{n}{j}-\binom{n}{j-1}$ for $j > 0$ and 1 for $j = 0$. See this paper for a proof.

• The Erdős-Ko-Rado theorem states that the independence number of the Kneser graph KG_n,k is

  $\alpha(KG_{n,k})=\binom{n-1}{k-1}$

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 ≥ 3k", Journal of Combinatorial Theory, Series B 80 (1): 69–79, doi:10.1006/jctb.2000.1969, MR 1778200.
• 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.
• 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", MathWorld.
• Weisstein, Eric W., "Odd Graph", MathWorld.