Editing Cube (section)

=== As a graph ===
{{main|Hypercube graph}}
[[File:Cube skeleton.svg|upright=0.8|thumb|The graph of a cube]]

According to [[Steinitz's theorem]], the [[Graph (discrete mathematics)|graph]] can be represented as the [[Skeleton (topology)|skeleton]] of a polyhedron; roughly speaking, a framework of a polyhedron. Such a graph has two properties: [[Planar graph|planar]] (the edges of a graph are connected to every vertex without crossing other edges), and [[k-vertex-connected graph|3-connected]] (whenever a graph with more than three vertices, and two of the vertices are removed, the edges remain connected).{{r|grunbaum-2003|ziegler}} The skeleton of a cube can be represented as the graph, and it is called the '''cubical graph''', a [[Platonic graph]]. It has the same number of vertices and edges as the cube, twelve vertices and eight edges.{{r|rudolph}} The cubical graph is also classified as a [[prism graph]], resembling the skeleton of a cuboid.{{r|ps}}

The cubical graph is a special case of [[hypercube graph]] or {{nowrap|1=<math>n</math>-}}cube&mdash;denoted as <math> Q_n </math>&mdash;because it can be constructed by using the operation known as the [[Cartesian product of graphs]]: it involves two graphs connecting the pair of vertices with an edge to form a new graph.{{r|hh}} In the case of the cubical graph, it is the product of two <math> Q_2 </math>; roughly speaking, it is a graph resembling a square. In other words, the cubical graph is constructed by connecting each vertex of two squares with an edge. Notationally, the cubical graph is <math> Q_3 </math>.{{r|cz}} Like any hypercube graph, it has a [[Cycle (graph theory)|cycle]] visits [[Hamiltonian path|every vertex exactly once]],{{r|ly}} and it is also an example of a [[unit distance graph]].{{r|hp}}

The cubical graph is [[bipartite graph|bipartite]], meaning every [[Independent set (graph theory)|independent set]] of four vertices can be [[Disjoint set|disjoint]] and the edges connected in those sets.{{r|berman-graph}} However, every vertex in one set cannot connect all vertices in the second, so this bipartite graph is not [[complete bipartite graph|complete]].{{r|aw}} It is an example of both [[crown graph]] and [[bipartite Kneser graph]].{{r|kl|berman-graph}}