# Kneser graph

Kneser graph
The Kneser graph KG5,2,
isomorphic to the Petersen graph
Named afterMartin Kneser
Vertices${\displaystyle \textstyle {\binom {n}{k}}}$
Edges${\displaystyle \textstyle {\binom {n}{k}}{\binom {n-k}{k}}/2}$
Chromatic number${\displaystyle \left\{{\begin{array}{ll}n-2k+2&n\geq 2k\\1&{\text{otherwise}}\end{array}}\right.}$
Properties${\displaystyle \textstyle {\binom {n-k}{k}}}$-regular
arc-transitive
NotationKGn,k, K(n,k)
Table of graphs and parameters

In graph theory, the Kneser graph KGn,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 KGn,1.

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

The Kneser graph KG2n − 1,n − 1 is known as the odd graph On; the odd graph O3 = KG5,2 is isomorphic to the Petersen graph.

## Properties

• The Kneser graph is vertex transitive and edge transitive. Each vertex has exactly ${\displaystyle \textstyle {\binom {n-k}{k}}}$ 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 KGn,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 ${\displaystyle n\geq (3k+1+{\sqrt {5k^{2}-2k+1}})/2}$, the Kneser graph KGn,k always contains a Hamiltonian cycle (Chen 2003). Note that ${\displaystyle (3k+1+{\sqrt {5k^{2}-2k+1}})/2<2.62k+1}$, so this condition is satisfied if ${\displaystyle n\geq 2.62k+1}$. It is also known that all Kneser graphs KGn,k with ${\displaystyle n=2k+2^{a}}$ for any integers ${\displaystyle k\geq 1}$ and ${\displaystyle a\geq 0}$ have a Hamiltonian cycle (Mütze, Nummenpalo & Walczak 2018). In particular, the odd graph On has a Hamiltonian cycle for all n ≥ 4. Moreover, 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, when n < ck it does not contain cliques of size c, whereas it does contain such cliques when nck. 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 KGn,k is
${\displaystyle \left\lceil {\frac {k-1}{n-2k}}\right\rceil +1}$ (Valencia-Pabon & Vera 2005).
• The graph spectrum of the Kneser graph KGn,k is given as follows:
For ${\displaystyle j=0,...,k}$, the eigenvalue ${\displaystyle \lambda _{j}=(-1)^{j}{\binom {n-k-j}{k-j}}}$ occurs with multiplicity ${\displaystyle {\binom {n}{j}}-{\binom {n}{j-1}}}$ for ${\displaystyle j>0}$ and 1 for ${\displaystyle j=0}$. See this paper for a proof.

## Related graphs

The Johnson graph Jn,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 KGn,2. Johnson graphs are closely related to the Johnson scheme, both of which are named after Selmer M. Johnson.

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

The bipartite Kneser graph Hn,k has as vertices the sets of k and nk 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 ${\displaystyle \textstyle {\binom {n-k}{k}}}$. The bipartite Kneser graph can be formed as a bipartite double cover of KGn,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 H5,2 is the Desargues graph and the bipartite Kneser graph Hn,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, 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..
• 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.