Hypercube Graph

The -hypercube graph, also called the -cube graph and commonly denoted or , is the graph, for a nonnegative integer , whose vertices are the binary strings where or 1 and two vertices are adjacent iff the symbols differ in exactly one coordinate.

For , each vertex of has vertex degree . The handshaking lemma therefore gives edges. Including the singleton graph , the edge counts are 0, 1, 4, 12, 32, 80, ... (OEIS A001787).

The graph of the -hypercube is given by the graph Cartesian product of path graphs . The hypercube graph is isomorphic to the Hasse diagram for the Boolean algebra of subsets of an -element set. is also isomorphic to the simplex graph of the complete graph (Alikhani and Ghanbari 2024).

The locally twisted cube graph () modifies the recursive perfect matching between two copies of the preceding graph. It retains the vertices and edges of , while reducing the graph diameter from to approximately (Yang et al. 2005). Unlike hypercube graphs, these graphs are not bipartite for .

The above figures show orthographic projections of some small -hypercube graphs using the first two of each vertex's set of coordinates. Note that above is a projection of the usual cube looking along a space diagonal so that the top and bottom vertices coincide, and hence only seven of the cube's eight vertices are visible. In addition, three of the central edges connect to the upper vertex, while the other three connect to the lower vertex.

Hypercube graphs may be computed in the Wolfram Language using the command HypercubeGraph[n], and precomputed properties of hypercube graphs are implemented in the Wolfram Language as GraphData[{"Hypercube", n}].

Special cases are summarized in the following table.

0	singleton graph
1	path graph
2	square graph
3	cubical graph
4	tesseract graph

For , hypercube graphs are Hamiltonian, and any Hamiltonian cycle of a labeled hypercube graph defines a cyclic binary Gray code (Skiena 1990, p. 149; Mütze 2024). The numbers of Hamiltonian cycles in hypercube graphs were considered by Dixon and Goodman (1975).

Hypercube graphs are also graceful (Maheo 1980, Kotzig 1981, Gallian 2026), antipodal, and (most likely) dominating unique.

The numbers of (directed) Hamiltonian paths whose endpoints are nonadjacent on an -hypercube graph for , 2, ... are 0, 0, 48, 48384, 129480729600, ... (OEIS A006070; extending the result of Gardner 1986, pp. 23-24). The numbers of (directed) Hamiltonian cycles for , 3, ... are 2, 12, 2688, 1813091520, ... (Harary et al. 1988; OEIS A003042). For , the graph has no Hamiltonian cycle under the convention that a graph cycle has at least three vertices.

Closed formulas for the numbers of cycles of length in are given by for odd and

			(1)
			(2)
			(3)
			(4)

(E. Weisstein, Nov. 16, 2014 and Apr. 19, 2023).

Hypercube graphs are distance-transitive, and therefore also distance-regular.

Alspach (2010) reports that Ringel showed in 1954 that the hypercube graphs admit Hamilton decompositions whenever is a power of 2. Alspach et al. (1990) showed that every for admits a Hamilton decomposition.

For , the hypercube graphs are also unit-distance (Gerbracht 2008), as illustrated above for the first few hypercube graphs. This follows by induction starting with a single unit-length edge. Given a unit-distance embedding of , choose a unit translation vector that is not the difference of any two existing vertices. There are only finitely many forbidden vectors, so such a choice exists. The original embedding and its translate have disjoint vertex sets. Joining corresponding vertices by unit-length edges constructs .

Determining the domination number is intrinsically difficult (Azarija et al. 2017) and exact values are known for (Östergård and Blass 2001, Bertolo et al. 2004) and the infinite families and for (Azarija et al. 2017, Theorem 3). Azarija et al. (2017) showed that domination and total domination numbers of the hypercube graph are related by .

is planar for , so has graph crossing number for . Eggleton and Guy (1970) claimed to have discovered an upper bound for the graph crossing number of for , where

			(5)
			(6)

The first few values for , 4, ... are 0, 8, 56, 352, 1760, 8192, 35712, ... (OEIS A307813).

An error was subsequently found, but Erdős and Guy (1973) then conjectured that not only was the original bound correct (though not yet proved), but that (Clancy et al. 2020, pp. 258-259). While it is known that , exact values for larger are not known (Clancy et al. 2020, p. 259). However, upper bounds are directly computable using QuickCross (Haythorpe) which correspond to the Eggleton and Guy values for (E. Weisstein, Apr. 30, 2019). In addition, the Erdős and Guy (1973) conjecture has now been refuted since it is known that (Clancy et al. 2020, p. 259).

The coil-in-the-box problem asks for the maximum length of an induced cycle in . The induced condition excludes edges between nonconsecutive vertices of the cycle.

The geodesic transversal number is the minimum cardinality of a vertex set meeting every inclusion-maximal graph geodesic. Chen et al. (2026) proved

(7)

for , where and denote the ceiling function and floor function.