Editing Complete graph (section)

==Geometry and topology==
[[File:Csaszar_polyhedron_3D_model.svg|thumb|100px|Interactive [[Csaszar polyhedron]] model with vertices representing nodes. In [http://upload.wikimedia.org/wikipedia/commons/d/db/Csaszar_polyhedron_3D_model.svg the SVG image], move the mouse to rotate it.<ref>Ákos Császár, [http://www.diale.org/pdf/csaszar.pdf ''A Polyhedron Without Diagonals.''] {{Webarchive|url=https://web.archive.org/web/20170918064243/http://www.diale.org/pdf/csaszar.pdf |date=2017-09-18 }}, Bolyai Institute, University of Szeged, 1949</ref>]]

A complete graph with {{mvar|n}} nodes represents the edges of an {{math|(''n'' − 1)}}-[[simplex]].  Geometrically {{math|''K''{{sub|3}}}} forms the edge set of a [[triangle]], {{math|''K''{{sub|4}}}} a [[tetrahedron]], etc.  The [[Császár polyhedron]], a nonconvex polyhedron with the topology of a [[torus]], has the complete graph {{math|''K''{{sub|7}}}} as its [[skeleton (topology)|skeleton]].<ref>{{citation
  | last = Gardner
  | first = Martin
  | authorlink = Martin Gardner
  | title = Time Travel and Other Mathematical Bewilderments
  | publisher = W. H. Freeman and Company
  | year = 1988
  | pages = 140
  | bibcode = 1988ttom.book.....G
  | isbn = 0-7167-1924-X
  | url = https://archive.org/details/timetravelotherm0000gard_u0o1/mode/2up
  }}</ref> Every [[neighborly polytope]] in four or more dimensions also has a complete skeleton.

{{math|''K''{{sub|1}}}} through {{math|''K''{{sub|4}}}} are all [[planar graph]]s.  However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph {{math|''K''{{sub|5}}}} plays a key role in the characterizations of planar graphs: by [[Kuratowski's theorem]], a graph is planar if and only if it contains neither {{math|''K''{{sub|5}}}} nor the [[complete bipartite graph]] {{math|''K''{{sub|3,3}}}} as a subdivision, and by [[Wagner's theorem]] the same result holds for [[graph minor]]s in place of subdivisions.  As part of the [[Petersen family]], {{math|''K''{{sub|6}}}} plays a similar role as one of the [[forbidden minor]]s for [[linkless embedding]].<ref>{{citation
 | last1 = Robertson | first1 = Neil | author1-link = Neil Robertson (mathematician)
 | last2 = Seymour | first2 = P. D. | author2-link = Paul Seymour (mathematician)
 | last3 = Thomas | first3 = Robin | author3-link = Robin Thomas (mathematician)
 | doi = 10.1090/S0273-0979-1993-00335-5
 | arxiv = math/9301216 | mr = 1164063
 | issue = 1
 | journal = Bulletin of the American Mathematical Society
 | pages = 84–89
 | title = Linkless embeddings of graphs in 3-space
 | volume = 28
 | year = 1993
| s2cid = 1110662 }}.</ref> In other words, and as Conway and Gordon<ref>{{cite journal |author-link1=J. H. Conway|author-link2=Cameron Gordon (mathematician)|author1=Conway, J. H. |author2=Cameron Gordon|title=Knots and Links in Spatial Graphs |journal=[[Journal of Graph Theory]] |volume=7 |issue=4 |pages=445–453 |year=1983 |doi=10.1002/jgt.3190070410}}</ref> proved, every embedding of {{math|''K''{{sub|6}}}} into three-dimensional space is intrinsically linked, with at least one pair of linked triangles.  Conway and Gordon also showed that any three-dimensional embedding of {{math|''K''{{sub|7}}}} contains a [[Hamiltonian cycle]] that is embedded in space as a [[Knot (mathematics)|nontrivial knot]].