 Hypercube graph

Hypercube graph
The hypercube graph Q_{4}Vertices 2^{n} Edges 2^{n−1}n Diameter n Girth 4 if n≥2 Chromatic number 2 Spectral Gap Properties Symmetric
Distance regular
Unit distance
HamiltonianNotation Q_{n} v · graph theory, the hypercube graph Q_{n} is a regular graph with 2^{n} vertices, which correspond to the subsets of a set with n elements. Two vertices labelled by subsets W and B are joined by an edge if and only if W can be obtained from B by adding or removing a single element. Each vertex of Q_{n} is incident to exactly n edges (that is, Q_{n} is nregular), so, by the handshaking lemma the total number of edges is 2^{n−1}n.
The name comes from the fact that the hypercube graph is the onedimensional skeleton of the geometric hypercube.
Hypercube graphs should not be confused with cubic graphs, which are graphs that are 3regular. The only hypercube that is a cubic graph is Q_{3}.
Contents
Construction
The hypercube graph Q_{n} may be constructed using 2^{n} vertices labeled 0,1,2,...,2^{n}1 and connecting two vertices whenever their Hamming distance equals 1. Additionally Q_{n+1} may be constructed using two hypercubes Q_{n} and joining equivalent vertices together as shown in the above picture. The joining edges are called the (n+1)th dimension of Q_{n}. Each dimension of Q_{n} forms a perfect matching. Another definition of Q_{n} is the Cartesian product of n twovertex complete graphs K_{2}.
Properties
Hamiltonicity
It is a well known fact that every hypercube Q_{n} is Hamiltonian for n > 1. Additionally, a Hamiltonian path exists between vertices u,v iff u,v are part of different bipartitions. Both facts are easy to prove using the principle of induction and the inductive construction of hypercube graphs.
Hamiltonicity of the hypercube is tightly related to the theory of Gray codes. More precisely there is a bijective correspondence between the set of nbit cyclic Gray codes and the set of Hamiltonian cycles in the hypercube Q_{n}.^{[1]} An analogous property holds for acyclic nbit Gray codes and Hamiltonian paths.
A lesser known fact is that every perfect matching in the hypercube extends to a Hamiltonian cycle.^{[2]} The question whether every matching extends to a Hamiltonian cycle remains an open problem.^{[3]}
Bipartiteness
Every hypercube is a bipartite graph. If the function s(x) represents the number of set bits in the binary representation of x, then (assuming the vertices are labeled 0,1,...,2^{n}1 as in the definition)
defines a proper two coloring of Q_{n}.
Other properties
The hypercube graph Q_{n} (n > 1) :
 is the Hasse diagram of a finite Boolean algebra.
 is a median graph. Every median graph is an isometric subgraph of a hypercube, and can be formed as a retraction of a hypercube.
 has more than 2^{2n2} perfect matchings. (this is another consequence that follows easily from the inductive construction.)
 is arc transitive and symmetric. The symmetries of hypercube graphs can be represented as signed permutations.
 contains all the cycles of length 4,6,...,2^{n}2,2^{n} and is thus a bipanicyclic graph.
 can be drawn as a unit distance graph in the Euclidean plane by choosing a unit vector for each set element and placing each vertex corresponding to a set S at the sum of the vectors in S.
 is a nvertexconnected graph, by Balinski's theorem
 The family Q_{n} (n > 1) is a Lévy family of graphs
 The achromatic number of Q_{n} is known to be proportional to , but the constant of proportionality is not known precisely.^{[4]}
 The bandwidth of Q_{n} is exactly .^{[5]}
 The eigenvalues of the adjacency matrix are (n,n+2,...,2,0,2,...,n2,n) and the eigenvalues of its Laplacian are (0,2,...,2n). The kth eigenvalue has multiplicity in both cases.
 The isoperimetric number is h(G)=1
Problems
The problem of finding the longest path or cycle that is an induced subgraph of a given hypercube graph is known as the snakeinthebox problem.
Szymanski's conjecture concerns the suitability of a hypercube as an network topology for communications. It states that, no matter how one chooses a permutation connecting each hypercube vertex to another vertex with which it should be connected, there is always a way to connect these pairs of vertices by paths that do not share any directed edge.^{[6]}
Examples
The graph Q_{0} consists of a single vertex, while Q_{1} is the complete graph on two vertices and Q_{2} is a cycle of length 4.
The graph Q_{3} is the 1skeleton of a cube, a planar graph with eight vertices and twelve edges.
The graph Q_{4} is the Levi graph of the Möbius configuration.
See also
 Cubeconnected cycles
 Fibonacci cube
 Folded cube graph
Notes
 ^ Mills, W. H. (1963), "Some complete cycles on the ncube", Proceedings of the American Mathematical Society (American Mathematical Society) 14 (4): 640–643, doi:10.2307/2034292, JSTOR 2034292 .
 ^ Fink, J. (2007), "Perfect matchings extend to Hamiltonian cycles in hypercubes", Journal of Combinatorial Theory, Series B 97 (6): 1074–1076, doi:10.1016/j.jctb.2007.02.007 .
 ^ Ruskey, F. and Savage, C. Matchings extend to Hamiltonian cycles in hypercubes on Open Problem Garden. 2007.
 ^ Roichman, Y. (2000), "On the Achromatic Number of Hypercubes", Journal of Combinatorial Theory, Series B 79 (2): 177–182, doi:10.1006/jctb.2000.1955 .
 ^ Optimal Numberings and Isoperimetric Problems on Graphs, L.H. Harper, Journal of Combinatorial Theory, 1, 385–393, doi:10.1016/S00219800(66)800595
 ^ Szymanski, Ted H. (1989), "On the Permutation Capability of a CircuitSwitched Hypercube", Proc. Internat. Conf. on Parallel Processing, 1, Silver Spring, MD: IEEE Computer Society Press, pp. 103–110 .
References
 Harary, F.; Hayes, J. P.; Wu, H.J. (1988), "A survey of the theory of hypercube graphs", Computers & Mathematics with Applications 15 (4): 277–289, doi:10.1016/08981221(88)902131 .
Categories: Parametric families of graphs
 Regular graphs
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Hypercube graph
 Hypercube graph

Hypercube graph
The hypercube graph Q_{4}Vertices 2^{n} Edges 2^{n−1}n Diameter n Girth 4 if n≥2 Chromatic number 2 Spectral Gap Properties Symmetric
Distance regular
Unit distance
HamiltonianNotation Q_{n}