Hypercube graph explained
| Hypercube graph |
| Vertices: |
|
| Edges: |
|
| Automorphisms: |
|
| Chromatic Number: | 2 |
| Girth: | 4 if
|
| Diameter: |
|
| Spectrum: | \{(n-2k)\binom{n{k}};k=0,\ldots,n\}
|
| Notation: |
|
In graph theory, the hypercube graph
is the
edge graph of the
-dimensional
hypercube, that is, it is the graph formed from the vertices and edges of the hypercube. For instance, the cube graph
is the graph formed by the 8 vertices and 12 edges of a three-dimensional cube.
has
vertices,
edges, and is a
regular graph with
edges touching each vertex.
The hypercube graph
may also be constructed by creating a vertex for each
subset of an
-element set, with two vertices adjacent when their subsets differ in a single element, or by creating a vertex for each
-digit
binary number, with two vertices adjacent when their binary representations differ in a single digit. It is the
-fold
Cartesian product of the two-vertex
complete graph, and may be decomposed into two copies of
connected to each other by a
perfect matching.
Hypercube graphs should not be confused with cubic graphs, which are graphs that have exactly three edges touching each vertex. The only hypercube graph
that is a cubic graph is the cubical graph
.
Construction
The hypercube graph
may be constructed from the family of
subsets of a
set with
elements, by making a vertex for each possible subset and joining two vertices by an edge whenever the corresponding subsets differ in a single element. Equivalently, it may be constructed using
vertices labeled with
-bit
binary numbers and connecting two vertices by an edge whenever the
Hamming distance of their labels is one. These two constructions are closely related: a binary number may be interpreted as a set (the set of positions where it has a nonzero digit), and two such sets differ in a single element whenever the corresponding two binary numbers have Hamming distance one.
Alternatively,
may be constructed from the
disjoint union of two hypercubes
, by adding an edge from each vertex in one copy of
to the corresponding vertex in the other copy, as shown in the figure. The joining edges form a
perfect matching.
The above construction gives a recursive algorithm for constructing the adjacency matrix of a hypercube,
. Copying is done via the
Kronecker product
, so that the two copies of
have an adjacency matrix
,where
is the
identity matrix. Meanwhile the joining edges have an adjacency matrix
. The sum of these two terms gives a recursive function for the adjacency matrix of a hypercube:
Another construction of
is the
Cartesian product of
two-vertex complete graphs
. More generally the Cartesian product of copies of a complete graph is called a
Hamming graph; the hypercube graphs are examples of Hamming graphs.
Examples
The graph
consists of a single vertex, while
is the
complete graph on two vertices.
is a
cycle of
The graph
is the
1-skeleton of a
cube and is a planar graph with eight
vertices and twelve
edges.
The graph
is the
Levi graph of the
Möbius configuration. It is also the
knight's graph for a
toroidal
chessboard.
[1] Properties
Bipartiteness
Every hypercube graph is bipartite: it can be colored with only two colors. The two colors of this coloring may be found from the subset construction of hypercube graphs, by giving one color to the subsets that have an even number of elements and the other color to the subsets with an odd number of elements.
Hamiltonicity
Every hypercube
with
has a
Hamiltonian cycle, a cycle that visits each vertex exactly once. Additionally, a
Hamiltonian path exists between two vertices
and
if and only if they have different colors in a 2-coloring of the graph. Both facts are easy to prove using the principle of
induction on the dimension of the hypercube, and the construction of the hypercube graph by joining two smaller hypercubes with a matching.
Hamiltonicity of the hypercube is tightly related to the theory of Gray codes. More precisely there is a bijective correspondence between the set of
-bit cyclic Gray codes and the set of Hamiltonian cycles in the An analogous property holds for acyclic
-bit 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]
Other properties
The hypercube graph
(for
) :
perfect matchings. (this is another consequence that follows easily from the inductive construction.)
and is thus a
bipancyclic graph.
- can be drawn as a unit distance graph in the Euclidean plane by using the construction of the hypercube graph from subsets of a set of
elements, choosing a distinct
unit vector for each set element, and placing the vertex corresponding to the set
at the sum of the vectors in
.
. For larger values of
, the hypercube has
}
spanning trees.
\binom{i}{\lfloori/2\rfloor}
.
[4]
, but the constant of proportionality is not known precisely.
[5]
\{-n,-n+2,-n+4,...,n-4,n-2,n\}
and as the eigenvalues of its Laplacian matrix the numbers
. The
th eigenvalue has multiplicity
in both cases.
.
The family
for all
is a
Lévy family of graphs.
Problems
The problem of finding the longest path or cycle that is an induced subgraph of a given hypercube graph is known as the snake-in-the-box problem.
Szymanski's conjecture concerns the suitability of a hypercube as a 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]
See also
References
Notes and References
- .
- .
- Ruskey, F. and Savage, C. Matchings extend to Hamiltonian cycles in hypercubes on Open Problem Garden. 2007.
- Optimal Numberings and Isoperimetric Problems on Graphs, L.H. Harper, Journal of Combinatorial Theory, 1, 385 - 393,
- .
- .
