Hypercube Graph

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

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 on elements. is also isomorphic to the simplex graph of the complete graph (Alikhani and Ghanbari 2024).

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

All hypercube graphs are Hamiltonian, and any Hamiltonian cycle of a labeled hypercube graph defines a binary reflected Gray code (Skiena 1990, p. 149; Mütze 2024). Hypercube graphs are also graceful (Maheo 1980, Kotzig 1981, Gallian 2018), antipodal, and (most likely) dominating unique.

The numbers of (directed) Hamiltonian paths on an -hypercube graph for , 2, ... are 0, 0, 48, 48384, 129480729600, ... (OEIS A006070; extending the result of Gardner 1986, pp. 23-24), while the numbers of (directed) Hamiltonian cycles are 0, 2, 12, 2688, 1813091520, ... (Harary et al. 1988; OEIS A091299).

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.

In 1954, Ringel showed that the hypercube graphs admit Hamilton decompositions whenever is a power of 2 (Alspach 2010). 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 can be established by induction for the -hypercube graph by starting with the unit-distance embedding of the square graph, translating the embedding by one unit in a direction not chosen in any of the steps before (only finitely many unit translation vectors have been used, so there must be a direction not used before), connecting the vertices in the translate with the corresponding vertices in the original one, and repeating until the -hypercube graph has been constructed.

Determining the domination number is intrinsically difficult (Azarija et al. 2017) and as of April 2018, values are known only up to (Östergård and Blass 2001, Bertolo et al. 2004). 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 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. 2019). While it is known that , exact values for larger are not known (Clancy et al. 2019). 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. 2019).