In graph theory, the hypercube graph is the edge graph-dimensional hypercube, that is, it is the graph formed from the vertices and edges of the hypercube. For instance, the cube graph vertices edges, and is a regular graph edges touching each vertex.

may also be constructed by creating a vertex for each subset-digit binary number-fold Cartesian product of the two-vertex complete graph connected to each other by a perfect matching.

Hypercube graphs should not be confused with cubic graphs.

Construction

may be constructed from the family of subsets of a set-bit binary numbers and connecting two vertices by an edge whenever the Hamming distance of their labels is one. These two constructions are closely related: a binary number may be interpreted as a set (the set of positions where it has a nonzero digit), and two such sets differ in a single element whenever the corresponding two binary numbers have Hamming distance one.

may be constructed from the disjoint union to the corresponding vertex in the other copy, as shown in the figure. The joining edges form a perfect matching.

The above construction gives a recursive algorithm for constructing the adjacency matrix. Copying is done via the Kronecker product identity matrix is the Cartesian product. More generally the Cartesian product of copies of a complete graph is called a Hamming graph; the hypercube graphs are examples of Hamming graphs.

Examples

is the complete graph on two vertices.

is a cycle of

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

is the Levi graph of the Möbius configuration. It is also the knight's graph for a toroidal chessboard.

Properties

Bipartiteness

Every hypercube graph is bipartite: it can be colored with only two colors. The two colors of this coloring may be found from the subset construction of hypercube graphs, by giving one color to the subsets that have an even number of elements and the other color to the subsets with an odd number of elements.

Hamiltonicity

has a Hamiltonian cycle, a cycle that visits each vertex exactly once. Additionally, a Hamiltonian path if and only if they have different colors in a 2-coloring of the graph. Both facts are easy to prove using the principle of induction on the dimension of the hypercube, and the construction of the hypercube graph by joining two smaller hypercubes with a matching.

Hamiltonicity of the hypercube is tightly related to the theory of Gray codes. More precisely there is a bijective-bit cyclic Gray codes and the set of Hamiltonian cycles in the -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.

Other properties

) :

• 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.

• 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.

• and is thus a bipancyclic graph.

• can be drawn as a unit distance graph elements, choosing a distinct unit vector.

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

• is planar (can be drawn, the hypercube has

• spanning trees.

• has bandwidth.

• has achromatic number, but the constant of proportionality is not known precisely.

• has as the eigenvalues of its adjacency matrix in both cases.

• has isoperimetric number.

is a Lévy family of graphs.

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 a 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.

See also

• de Bruijn graph

• Cube-connected cycles

• Fibonacci cube

• Folded cube graph

• Frankl–Rödl graph

• Halved cube graph

• Hypercube internetwork topology

• Partial cube

Notes

References

• .