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Hypercube Graph

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The n-hypercube graph, also called the n-cube graph and commonly denoted Q_n or 2^n, is the graph whose vertices are the 2^k symbols epsilon_1, ..., epsilon_n where epsilon_i=0 or 1 and two vertices are adjacent iff the symbols differ in exactly one coordinate.

The graph of the n-hypercube is given by the graph Cartesian product of path graphs P_2×... square P_2_()_(n). The n-hypercube graph is also isomorphic to the Hasse diagram for the Boolean algebra on n elements.

HypercubeGraphIsometricEmbeddings

The above figures show orthographic projections of some small n-hypercube graphs using the first two of each vertex's set of n coordinates. Note that Q_3 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.

nQ_n
0singleton graph K_1
1path graph P_2
2square graph C_4
3cubical graph
4tesseract graph

All hypercube graphs are Hamiltonian, and any Hamiltonian cycle of a labeled hypercube graph defines a Gray code (Skiena 1990, p. 149). Hypercube graphs are also graceful (Maheo 1980, Kotzig 1981, Gallian 2018).

The numbers of (directed) Hamiltonian paths on an n-hypercube graph for n=1, 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 c_k of k-graph cycles of Q_n are given by c_k=0 for k odd and

c_4=2^(n-3)(n-1)n
(1)
c_6=1/32^n(n-2)(n-1)n
(2)
c_8=2^(n-4)(n-2)(n-1)n(27n-79)
(3)

(E. Weisstein, Nov. 16, 2014).

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

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

HypercubeGraphUnitDistance

For n>=1, 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 n-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 n-hypercube graph has been constructed.

Determining the domination number gamma(Q_n) is intrinsically difficult (Azarija et al. 2017) and as of April 2018, values are known only up to n=9 (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 gamma_t(Q_(n+1))=2gamma(Q_n).

Q_n is planar for n<=3, so has graph crossing number cr(Q_n) for n<=3. Eggleton and Guy (1970) claimed to have discovered an upper bound for the graph crossing number of cr(Q_n)<=a(n) for n>=3, where

a(n)=5/(32)4^n-|_(n^2+1)/2_|2^(n-2)
(4)
=2^(n-5)(5·2^n-4n^2+2(-1)^n-2).
(5)

The first few values for n=3, 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 cr(Q_n)=a(n) (Clancy et al. 2019). While it is known that cr(Q_4)=8, exact values for larger n 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 n<=6 (E. Weisstein, Apr. 30, 2019). In addition, the Erdős and Guy (1973) conjecture has now been refuted since it is known that cr(Q_7)<=1744<a(7) (Clancy et al. 2019).

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