Толық граф теориясы

Graph in which every two vertices are adjacent

In the mathematical field of graph theory, a complete graph is a simple undirected graph in which every pair of distinct vertices is connected by a unique edge. A complete digraph is a directed graph in which every pair of distinct vertices is connected by a pair of unique edges (one in each direction). Graph theory itself is typically dated as beginning with Leonhard Euler's 1736 work on the Seven Bridges of Königsberg. However, drawings of complete graphs, with their vertices placed on the points of a regular polygon, had already appeared in the 13th century, in the work of Ramon Llull. Such a drawing is sometimes referred to as a mystic rose.

A complete graph with n nodes represents the edges of an (n – 1) 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.