The complete graph on vertices is denoted by. Some sources claim that the letter K in this notation stands for the German word komplett, but the German name for a complete graph, vollständiger Graph, does not contain the letter K, and other sources state that the notation honors the contributions of Kazimierz Kuratowski to graph theory. Kn has edges, and is a regular graph of degree. All complete graphs are their own maximal cliques. They are maximally connected as the only vertex cut which disconnects the graph is the complete set of vertices. The complement graph of a complete graph is an empty graph. If the edges of a complete graph are each given an orientation, the resulting directed graph is called a tournament. Kn can be decomposed into trees Ti such that Ti has vertices. Ringel's conjecture asks if the complete graph K2n+1 can be decomposed into copies of any tree with edges. This is known to be true for sufficiently large. The number of matchings of the complete graphs are given by the telephone numbers These numbers give the largest possible value of the Hosoya index for an n-vertex graph. The number of perfect matchings of the complete graph Kn is given by the double factorial !!. The crossing numbers up to K27 are known, with K28 requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project. Rectilinear Crossing numbers for Kn are
Geometry and topology
A complete graph with nodes represents the edges of an -simplex. Geometrically forms the edge set of a triangle, a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph as its skeleton. Every neighborly polytope in four or more dimensions also has a complete skeleton. through are all planar graphs. However, every planar drawing of a complete graph with five or more vertices must contain a crossing, and the nonplanar complete graph 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 nor the complete bipartite graph as a subdivision, and by Wagner's theorem the same result holds for graph minors in place of subdivisions. As part of the Petersen family, plays a similar role as one of the forbidden minors for linkless embedding. In other words, and as Conway and Gordon proved, every embedding of 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 contains a Hamiltonian cycle that is embedded in space as a nontrivial knot.
Examples
Complete graphs on vertices, for between 1 and 12, are shown below along with the numbers of edges:
