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 also isomorphic to the Hasse diagram for the Boolean algebra on  elements.

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 Gray code (Skiena 1990, p. 149). Hypercube graphs are also graceful (Maheo 1980, Kotzig 1981, Gallian 2018). Hypercube graphs are also antipodal.

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 -graph cycles of  are given by  for  odd and

			(1)
			(2)
			(3)

(E. Weisstein, Nov. 16, 2014).

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

			(4)
			(5)

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).

REFERENCES

Alspach, B. "Three Hamilton Decomposition Problems." University of Western Australia. May 11, 2010.

 http://symomega.files.wordpress.com/2010/05/talk8.pdf.Alspach, B.; Bermond, J.-C.; and Sotteau, D. "Decomposition Into Cycles. I. Hamilton Decompositions." In Proceedings of the NATO Advanced Research Workshop on Cycles and Rays: Basic Structures in Finite and Infinite Graphs held in Montreal, Quebec, May 3-9, 1987

 (Ed. G. Hahn, G. Sabidussi, and R. E. Woodrow). Dordrecht, Holland: Kluwer, pp. 9-18, 1990.

Azarija, J.; Henning, M. A.; and Klavžar, S. "(Total) Domination in Prisms." Electron. J. Combin. 24, No. 1, paper 1.19, 2017.

 http://www.combinatorics.org/ojs/index.php/eljc/article/view/v24i1p19.Bertolo, R.; Östergård, P. R. J.; and Weakley, W. D. "An Updated Table of Binary/ternary Mixed Covering Codes." J. Combin. Des. 12, 157-176, 2004.

Biggs, N. L. Algebraic Graph Theory, 2nd ed. Cambridge, England: Cambridge University Press, p. 161, 1993.

Clancy, K.; Haythorpe, M.; and Newcombe, A. "A Survey of Graphs with Known or Bounded Crossing Numbers." 15 Feb 2019.

 https://arxiv.org/pdf/1901.05155.pdf.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.

 https://www.combinatorics.org/ojs/index.php/eljc/article/view/DS6.Gardner, M. "The Binary Gray Code." In Knotted Doughnuts and Other Mathematical Entertainments. New York: W. H. Freeman, pp. 23-24, 1986.

Gerbracht, E. H.-A. "On the Unit Distance Embeddability of Connected Cubic Symmetric Graphs." Kolloquium über Kombinatorik. Magdeburg, Germany. Nov. 15, 2008.

Gross, J. T. and Yellen, J. Graph Theory and Its Applications. Boca Raton, FL: CRC Press, p. 14, 1999.

Harary, F.; Hayes, J. P.; and Wu, H.-J. "A Survey of the Theory of Hypercube Graphs." Comput. Math. Appl. 15, 277-289, 1988.

Haythorpe, M. "QuickCross--Crossing Number Problem." http://www.flinders.edu.au/science_engineering/csem/research/programs/flinders-hamiltonian-cycle-project/quickcross.cfm.Kotzig, A. "Decomposition of Complete Graphs Into Isomorphic Cubes." J. Combin. Th. 31, 292-296, 1981.

Maheo, M. "Strongly Graceful Graphs." Disc. Math. 29, 39-46, 1980.

Östergård, P. R. J. and Blass, U. "On the Size of Optimal Binary Codes of Length 9 and Covering Radius 1." IEEE Trans. Inform. Th. 47, 2556-2557, 2001.

Skiena, S. "Hypercubes." §4.2.5 in Implementing Discrete Mathematics: Combinatorics and Graph Theory with Mathematica. Reading, MA: Addison-Wesley, pp. 148-150, 1990.

Sloane, N. J. A. Sequences A006070, A091299, and A307813 in "The On-Line Encyclopedia of Integer Sequences."