The 
-hypercube
graph, also called the 
-cube graph and commonly denoted
or 
, is the graph whose vertices
are the 
binary strings 
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.
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 2018), 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.
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 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). While it is known
that 
,
exact values for larger 
are not known (Clancy et al. 2020). 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).
."
-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., Dynamic Survey DS6, Oct. 30, 2025.