Kneser graph

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

In graph theory, the Kneser graphK(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 1956.

Examples

Kneser graph O₄ = K(7, 3)

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 (see top right figure).

The Kneser graph O₄ = K(7, 3), visualized on the right.

Properties

Basic properties

The Kneser graph ${\displaystyle K(n,k)}$ 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. When ${\displaystyle k=2}$, the Kneser graph is a strongly regular graph, with parameters ${\displaystyle ({\tbinom {n}{2}},{\tbinom {n-2}{2}},{\tbinom {n-4}{2}},{\tbinom {n-3}{2}})}$. However, it is not strongly regular when ${\displaystyle k>2}$, 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 ${\displaystyle K(2k,k)}$ which is disconnected. More precisely, the connectivity of ${\displaystyle K(n,k)}$ is ${\displaystyle {\tbinom {n-k}{k}},}$ the same as the number of neighbors per vertex. ^[1]

Chromatic number

As Kneser ( 1956 ) conjectured, the chromatic number of the Kneser graph ${\displaystyle K(n,k)}$ for ${\displaystyle n\geq 2k}$ is exactly n − 2k + 2; for instance, the Petersen graph requires three colors in any proper coloring. This conjecture was proved in several ways.

• László Lovász proved this in 1978 using topological methods, ^[2] 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. ^[3]
• Joshua E. Greene won the 2002 the Morgan Prize for outstanding undergraduate research for his further simplified but still topological proof. ^[4]
• In 2004, Jiří Matoušek found a purely combinatorial proof. ^[5]

In contrast, the fractional chromatic number of these graphs is ${\displaystyle n/k}$. ^[6] When ${\displaystyle n<2k}$, ${\displaystyle K(n,k)}$ has no edges and its chromatic number is 1.

Hamiltonian cycles

It is well-known that the Petersen graph is not Hamiltonian, but it was long conjectured that this was the sole exception and that every other connected Kneser graph K(n, k) is Hamiltonian.

In 2003, Chen showed that the Kneser graph K(n, k) contains a Hamiltonian cycle if ^[7]

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

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 ${\displaystyle k}$ this condition is satisfied if

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

Around the same time, Shields showed (computationally) that, except the Petersen graph, all connected Kneser graphs K(n, k) with n ≤ 27 are Hamiltonian. ^[8]

In 2021, Mütze, Nummenpalo, and Walczak proved that the Kneser graph K(n, k) contains a Hamiltonian cycle if there exists a non-negative integer ${\displaystyle a}$ such that ${\displaystyle n=2k+2^{a}}$. ^[9] In particular, the odd graph O_n has a Hamiltonian cycle if n ≥ 4. Finally, in 2023, Merino, Mütze and Namrata completed the proof of the conjecture. ^[10]

Cliques

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 variable length. ^[11]

Diameter

The diameter of a connected Kneser graph K(n, k) is ^[12]

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

Spectrum

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. ^[13]

Independence number

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 graphK(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. ^[11] Thus K(n, k, 0) = K(n, k).

The bipartite Kneser graphH(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. ^[14] The bipartite Kneser graph H(5, 2) is the Desargues graph and the bipartite Kneser graph H(n, 1) is a crown graph.

References

Notes

1. Watkins (1970).
2. Lovász (1978).
3. Bárány (1978).
4. Greene (2002).
5. Matoušek (2004).
6. Godsil & Meagher (2015).
7. Chen (2003).
8. Shields (2004).
9. Mütze, Nummenpalo & Walczak (2021).
10. Merino, Mütze & Namrata (2023).
11. Denley (1997).
12. Valencia-Pabon & Vera (2005).
13. "Archived copy" (PDF). www.math.caltech.edu. Archived from the original (PDF) on 23 March 2012. Retrieved 9 August 2022.
14. Simpson (1991).

Works cited

• 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
• Godsil, Christopher; Meagher, Karen (2015), Erdős–Ko–Rado Theorems: Algebraic Approaches, Cambridge Studies in Advanced Mathematics, Cambridge University Press, p. 43, ISBN   9781107128446

• 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 (1956), "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, hdl: 20.500.11850/50671 , MR   2057690, S2CID   42583803
• Mütze, Torsten; Nummenpalo, Jerri; Walczak, Bartosz (2021) [STOC 2018], "Sparse Kneser graphs are Hamiltonian", Journal of the London Mathematical Society, New York, 103 (4): 912–919, arXiv: 1711.01636 , doi:10.1112/jlms.12406, MR   3826304
• Merino, Arturo; Mütze, Torsten; Namrata (2023), "Kneser graphs are Hamiltonian", Proceedings of the 55th Annual ACM Symposium on Theory of Computing, pp. 963–970, arXiv: 2212.03918 , doi: 10.1145/3564246.3585137
• 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, vol. 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 .