Hypercube graph

Hypercube graph
The hypercube graph Q4
Vertices	2n
Edges	2n−1n
Diameter	n
Girth	4 if n≥2
Chromatic number	2
Spectrum	$\{(n - 2 k)^{\binom{n}{k}}; k = 0, \ldots, n\}$
Properties	Symmetric Distance regular Unit distance Hamiltonian
Notation	Qn

In graph theory, the hypercube graph Q_n is a regular graph with 2ⁿ vertices, which correspond to the subsets of a set with n elements. Two vertices labelled by subsets W and B are joined by an edge if and only if W can be obtained from B by adding or removing a single element.

Each vertex of Q_n is incident to exactly n edges (that is, Q_n is n-regular), so, by the handshaking lemma the total number of edges is 2ⁿ⁻¹n.

The name comes from the fact that the hypercube graph is the one-dimensional skeleton of the geometric hypercube.

Hypercube graphs should not be confused with cubic graphs, which are graphs that are 3-regular. The only hypercube that is a cubic graph is Q₃.

Construction

An animation showing the construction of a hypercube graph of dimension 4

The hypercube graph Q_n may be constructed using 2ⁿ vertices labeled 0,1,2,...,2ⁿ-1 and connecting two vertices whenever their Hamming distance equals 1. Additionally Q_n+1 may be constructed using two hypercubes Q_n and joining equivalent vertices together as shown in the above picture. The joining edges are called the (n+1)th dimension of Q_n. Each dimension of Q_n forms a perfect matching. Another definition of Q_n is the Cartesian product of n two-vertex complete graphs K₂.

Inductive construction of Q₃ using two Q₂ hypercubes.

Properties

Hamiltonicity

It is a well known fact that every hypercube Q_n is Hamiltonian for n > 1. Additionally, a Hamiltonian path exists between vertices u,v iff u,v are part of different bipartitions. Both facts are easy to prove using the principle of induction and the inductive construction of hypercube graphs.

Hamiltonicity of the hypercube is tightly related to the theory of Gray codes. More precisely there is a bijective correspondence between the set of n-bit cyclic Gray codes and the set of Hamiltonian cycles in the hypercube Q_n. An analogous property holds for acyclic n-bit Gray codes and Hamiltonian paths.

A lesser known fact is that every perfect matching in the hypercube extends to a Hamiltonian cycle. The question whether every matching extends to a Hamiltonian cycle remains an open problem.

Bipartiteness

Every hypercube is a bipartite graph. If the function s(x) represents the number of set bits in the binary representation of x, then (assuming the vertices are labeled 0,1,...,2ⁿ-1 as in the definition)

$c(v) := s(v) \mod 2$

defines a proper two coloring of Q_n.

Other properties

The hypercube graph Q_n (n > 1) :

• is the Hasse diagram of a finite Boolean algebra.

• is a median graph. Every median graph is an isometric subgraph of a hypercube, and can be formed as a retraction of a hypercube.

• has more than 2^2n-2 perfect matchings. (this is another consequence that follows easily from the inductive construction.)

• is arc transitive and symmetric. The symmetries of hypercube graphs can be represented as signed permutations.

• contains all the cycles of length 4,6,...,2ⁿ-2,2ⁿ and is thus a bi-panicyclic graph.

• can be drawn as a unit distance graph in the Euclidean plane by choosing a unit vector for each set element and placing each vertex corresponding to a set S at the sum of the vectors in S.

• is a n-vertex-connected graph, by Balinski's theorem

• is planar (can be drawn with no crossings) if and only if n ≤ 3.

• The family Q_n (n > 1) is a Lévy family of graphs

• The achromatic number of Q_n is known to be proportional to $\sqrt{n2^n}$, but the constant of proportionality is not known precisely.

• The bandwidth of Q_n is exactly $\sum_{i=0}^n \binom{n}{\lfloor n/2\rfloor}$.

• The eigenvalues of the adjacency matrix are (-n,-n+2,-n+4,...,n-4,n-2,n) and the eigenvalues of its Laplacian are (0,2,...,2n). The k-th eigenvalue has multiplicity $\binom{n}{k}$ in both cases.

• The isoperimetric number is h(G)=1

Problems

The problem of finding the longest path or cycle that is an induced subgraph of a given hypercube graph is known as the snake-in-the-box problem.

Szymanski's conjecture concerns the suitability of a hypercube as an network topology for communications. It states that, no matter how one chooses a permutation connecting each hypercube vertex to another vertex with which it should be connected, there is always a way to connect these pairs of vertices by paths that do not share any directed edge.

Examples

The graph Q₀ consists of a single vertex, while Q₁ is the complete graph on two vertices and Q₂ is a cycle of length 4.

The graph Q₃ is the 1-skeleton of a cube, a planar graph with eight vertices and twelve edges.

The graph Q₄ is the Levi graph of the Möbius configuration.

See also

• Cube-connected cycles
• Fibonacci cube
• Folded cube graph

References

• Harary, F.; Hayes, J. P.; Wu, H.-J. (1988), "A survey of the theory of hypercube graphs", Computers & Mathematics with Applications 15 (4): 277–289, doi:10.1016/0898-1221(88)90213-1 .