Kneser graph

In graph theory, the Kneser graph K(n, k) (alternatively 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.

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

Kneser graph
The Kneser graph K(5, 2), isomorphic to the Petersen graph
Named after	Martin Kneser
Vertices	${\displaystyle {\binom {n}{k}}}$
Edges	${\displaystyle {\frac {1}{2}}{\binom {n}{k}}{\binom {n-k}{k}}}$
Chromatic number	${\displaystyle {\begin{cases}n-2k+2&n\geq 2k\\1&n<2k\end{cases}}}$
Properties	${\displaystyle {\tbinom {n-k}{k}}}$-regular arc-transitive
Notation	K(n, k), KGn,k.
Table of graphs and parameters

Examples

The Kneser graph K(n, 1) is the complete graph on n vertices.

The Kneser graph K(n, 2) is the complement of the line graph of the complete graph on n vertices.

The Kneser graph K(2n − 1, n − 1) is the odd graph O_n; in particular O₃ = K(5, 2) is the Petersen graph.

Properties

• The Kneser graph K(n, k) has ${\displaystyle {\tbinom {n}{k}}}$ vertices. Each vertex has exactly ${\displaystyle {\tbinom {n-k}{k}}}$ neighbors.

• The Kneser graph is vertex transitive and arc transitive. However, it 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.

• 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 ${\displaystyle {\tbinom {n-k}{k}},}$ the same as the number of neighbors per vertex (Watkins 1970).

• As Kneser (1955) conjectured, the chromatic number of the Kneser graph K(n, k) for ${\displaystyle n\geq 2k}$ 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.

• The Kneser graph K(n, k) contains a Hamiltonian cycle if (Chen 2003):

${\displaystyle n\geq {\frac {1}{2}}\left(3k+1+{\sqrt {5k^{2}-2k+1}}\right).}$

Since

${\displaystyle {\frac {1}{2}}\left(3k+1+{\sqrt {5k^{2}-2k+1}}\right)<\left({\frac {3+{\sqrt {5}}}{2}}\right)k+1,}$

holds for all k this condition is satisfied if

${\displaystyle n\geq \left({\frac {3+{\sqrt {5}}}{2}}\right)k+1\approx 2.62k+1.}$

• The Kneser graph K(n, k) contains a Hamiltonian cycle if there exists a non-negative integer a such that ${\displaystyle n=2k+2^{a}}$ (Mütze, Nummenpalo & Walczak 2018). In particular, the odd graph O_n has a Hamiltonian cycle if n ≥ 4.

• With the exception of the Petersen graph, all connected Kneser graphs K(n, k) with n ≤ 27 are Hamiltonian (Shields 2004).

• When n < 3k, the Kneser graph K(n, k) contains no triangles. More generally, when n < ck it does not contain cliques of size c, whereas it does contain such cliques when n ≥ ck. Moreover, 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 K(n, k) is (Valencia-Pabon & Vera 2005):

${\displaystyle \left\lceil {\frac {k-1}{n-2k}}\right\rceil +1.}$

• The spectrum of the Kneser graph K(n, k) consists of k + 1 distinct eigenvalues:

${\displaystyle \lambda _{j}=(-1)^{j}{\binom {n-k-j}{k-j}},\qquad j=0,\ldots ,k.}$

Moreover ${\displaystyle \lambda _{j}}$ occurs with multiplicity ${\displaystyle {\tbinom {n}{j}}-{\tbinom {n}{j-1}}}$ for ${\displaystyle j>0}$ and ${\displaystyle \lambda _{0}}$ has multiplicity 1. See this paper for a proof.

• The Erdős–Ko–Rado theorem states that the independence number of the Kneser graph K(n, k) for ${\displaystyle n\geq 2k}$ is

${\displaystyle \alpha (K(n,k))={\binom {n-1}{k-1}}.}$

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. The Johnson graph J(n, 2) is the complement of the Kneser graph K(n, 2). Johnson graphs are closely related to the Johnson scheme, both of which are named after Selmer M. Johnson.

The generalized Kneser graph K(n, k, s) has the same vertex set as the Kneser graph K(n, k), but connects two vertices whenever they correspond to sets that intersect in s or fewer items (Denley 1997). Thus K(n, k, 0) = K(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 ${\displaystyle {\tbinom {n-k}{k}}.}$ The bipartite Kneser graph can be formed as a bipartite double cover of K(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 (2003), "Triangle-free Hamiltonian Kneser graphs", Journal of Combinatorial Theory, Series B, 89 (1): 1–16, doi:10.1016/S0095-8956(03)00040-6, MR 1999733
• 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, JSTOR 3072460, MR 1941810
• Kneser, Martin (1955), "Aufgabe 360", Jahresbericht der Deutschen Mathematiker-Vereinigung, 58 (2): 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, hdl:10338.dmlcz/126050, 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
• Mütze, Torsten; Nummenpalo, Jerri; Walczak, Bartosz (2018), "Sparse Kneser graphs are Hamiltonian", STOC'18—Proceedings of the 50th Annual ACM SIGACT Symposium on Theory of Computing, New York: ACM, pp. 912–919, arXiv:1711.01636, MR 3826304
• Shields, Ian Beaumont (2004), Hamilton Cycle Heuristics in Hard Graphs, Ph.D. thesis, North Carolina State University, archived from the original on 2006-09-17, retrieved 2006-10-01
• 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
• Watkins, Mark E. (1970), "Connectivity of transitive graphs", Journal of Combinatorial Theory, 8: 23–29, doi:10.1016/S0021-9800(70)80005-9, MR 0266804

External links

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