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.
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). The numbers of Hamiltonian cycles in hypercube graphs were considered by Dickson and Goodman (1975).
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).
-Cube." Proc. Amer. Math. Soc. 50, 500-504,
1975.Eggleton, R. B. and Guy, R. K. "The Crossing Number
of the 
-Cube."
Not. Amer. Math. Soc. 17, 757, 1970.Erdős, P. and
Guy, R. K. "Crossing Number Problems." Amer. Math. Monthly 80,
52-58, 1973.Gallian, J. "Dynamic Survey of Graph Labeling."
Elec. J. Combin. DS6. Dec. 21, 2018.