Editing Octahedron (section)

=== Graph ===
[[File:Complex tripartite graph octahedron.svg|class=skin-invert-image|thumb|upright=0.8|The graph of a regular octahedron]]
The [[Skeleton (topology)|skeleton]] of a regular octahedron can be represented as a [[Graph (discrete mathematics)|graph]] according to [[Steinitz's theorem]], provided the graph is [[Planar graph|planar]]&mdash;its edges of a graph are connected to every vertex without crossing other edges&mdash;and [[k-vertex-connected graph|3-connected graph]]&mdash;its edges remain connected whenever two of more three vertices of a graph are removed.{{r|grunbaum-2003|ziegler}} Its graph called the '''octahedral graph''', a [[Platonic graph]].{{r|hs}}

The octahedral graph can be considered as [[Tripartite graph|complete tripartite graph]] <math> K_{2,2,2} </math>, a graph partitioned into three independent sets each consisting of two opposite vertices.{{r|negami}} More generally, it is a [[Turán graph]] <math> T_{6,3} </math>.

The octahedral graph is [[k-vertex-connected graph|4-connected]], meaning that it takes the removal of four vertices to disconnect the remaining vertices. It is one of only four 4-connected [[simplicial polytope|simplicial]] [[well-covered graph|well-covered]] polyhedra, meaning that all of the [[maximal independent set]]s of its vertices have the same size. The other three polyhedra with this property are the [[pentagonal dipyramid]], the [[snub disphenoid]], and an irregular polyhedron with 12 vertices and 20 triangular faces.{{r|fhnp}}