Complete graph  

Vertices  n 
Edges  
Radius  
Diameter  
Girth  
Automorphisms  n! (S _{n}) 
Chromatic number  n 
Chromatic index 

Spectrum  
Properties  
Notation  K_{n} 
Table of graphs and parameters 
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, appeared already in the 13th century, in the work of Ramon Llull.^{ [1] } Such a drawing is sometimes referred to as a mystic rose.^{ [2] }
The complete graph on n vertices is denoted by K_{n}. Some sources claim that the letter K in this notation stands for the German word komplett,^{ [3] } 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.^{ [4] }
K_{n} has n(n − 1)/2 edges (a triangular number), and is a regular graph of degree n− 1. 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.
K_{n} can be decomposed into n trees T_{i} such that T_{i} has i vertices.^{ [5] } Ringel's conjecture asks if the complete graph K_{2n+1} can be decomposed into copies of any tree with n edges.^{ [6] } This is known to be true for sufficiently large n.^{ [7] }^{ [8] }
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 nvertex graph.^{ [9] } The number of perfect matchings of the complete graph K_{n} (with n even) is given by the double factorial (n − 1)!!.^{ [10] }
The crossing numbers up to K_{27} are known, with K_{28} requiring either 7233 or 7234 crossings. Further values are collected by the Rectilinear Crossing Number project.^{ [11] } Rectilinear Crossing numbers for K_{n} are
A complete graph with n nodes represents the edges of an (n − 1)simplex. Geometrically K_{3} forms the edge set of a triangle, K_{4} a tetrahedron, etc. The Császár polyhedron, a nonconvex polyhedron with the topology of a torus, has the complete graph K_{7} as its skeleton. Every neighborly polytope in four or more dimensions also has a complete skeleton.
K_{1} through K_{4} 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 K_{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 K_{5} nor the complete bipartite graph K_{3,3} 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, K_{6} plays a similar role as one of the forbidden minors for linkless embedding.^{ [13] } In other words, and as Conway and Gordon^{ [14] } proved, every embedding of K_{6} into threedimensional space is intrinsically linked, with at least one pair of linked triangles. Conway and Gordon also showed that any threedimensional embedding of K_{7} contains a Hamiltonian cycle that is embedded in space as a nontrivial knot.
Complete graphs on n vertices, for n between 1 and 12, are shown below along with the numbers of edges:
K_{1}: 0  K_{2}: 1  K_{3}: 3  K_{4}: 6 

K_{5}: 10  K_{6}: 15  K_{7}: 21  K_{8}: 28 
K_{9}: 36  K_{10}: 45  K_{11}: 55  K_{12}: 66 
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
The classical mathematical puzzle known as the three utilities problem; the three cottages problem or sometimes water, gas and electricity can be stated as follows:
Suppose there are three cottages on a plane and each needs to be connected to the water, gas, and electricity companies. Without using a third dimension or sending any of the connections through another company or cottage, is there a way to make all nine connections without any of the lines crossing each other?
In graph theory, an outerplanar graph is a graph that has a planar drawing for which all vertices belong to the outer face of the drawing.
In graph theory, an undirected graph H is called a minor of the graph G if H can be formed from G by deleting edges and vertices and by contracting edges.
In the mathematical field of graph theory, a complete bipartite graph or biclique is a special kind of bipartite graph where every vertex of the first set is connected to every vertex of the second set.
In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into threedimensional Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a threedimensional analogue of the planar graphs. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding.
In the mathematical discipline of graph theory, the dual graph of a plane graph G is a graph that has a vertex for each face of G. The dual graph has an edge whenever two faces of G are separated from each other by an edge, and a selfloop when the same face appears on both sides of an edge. Thus, each edge e of G has a corresponding dual edge, whose endpoints are the dual vertices corresponding to the faces on either side of e. The definition of the dual depends on the choice of embedding of the graph G, so it is a property of plane graphs rather than planar graphs. For planar graphs generally, there may be multiple dual graphs, depending on the choice of planar embedding of the graph.
In mathematics, Fáry's theorem states that any simple planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn. The theorem is named after István Fáry, although it was proved independently by Klaus Wagner (1936), Fáry (1948), and Sherman K. Stein (1951).
In geometry, the Császár polyhedron is a nonconvex toroidal polyhedron with 14 triangular faces.
In graph theory, the crossing numbercr(G) of a graph G is the lowest number of edge crossings of a plane drawing of the graph G. For instance, a graph is planar if and only if its crossing number is zero. Determining the crossing number continues to be of great importance in graph drawing, as user studies have shown that drawing graphs with few crossings makes it easier for people to understand the drawing.
In polyhedral combinatorics, a branch of mathematics, Steinitz's theorem is a characterization of the undirected graphs formed by the edges and vertices of threedimensional convex polyhedra: they are exactly the (simple) 3vertexconnected planar graphs. That is, every convex polyhedron forms a 3connected planar graph, and every 3connected planar graph can be represented as the graph of a convex polyhedron. For this reason, the 3connected planar graphs are also known as polyhedral graphs. Branko Grünbaum has called this theorem "the most important and deepest known result on 3polytopes."
In the mathematics of graph drawing, Turán's brick factory problem asks for the minimum number of crossings in a drawing of a complete bipartite graph. The problem is named after Pál Turán, who formulated it while being forced to work in a brick factory during World War II.
In geometric graph theory, a branch of mathematics, a polyhedral graph is the undirected graph formed from the vertices and edges of a convex polyhedron. Alternatively, in purely graphtheoretic terms, the polyhedral graphs are the 3vertexconnected planar graphs.
In graph theory, a branch of mathematics, the Herschel graph is a bipartite undirected graph with 11 vertices and 18 edges, the smallest nonHamiltonian polyhedral graph. It is named after British astronomer Alexander Stewart Herschel.
Barnette's conjecture is an unsolved problem in graph theory, a branch of mathematics, concerning Hamiltonian cycles in graphs. It is named after David W. Barnette, a professor emeritus at the University of California, Davis; it states that every bipartite polyhedral graph with three edges per vertex has a Hamiltonian cycle.
In graph theory, the Petersen family is a set of seven undirected graphs that includes the Petersen graph and the complete graph K_{6}. The Petersen family is named after Danish mathematician Julius Petersen, the namesake of the Petersen graph.
In graph theory, a branch of mathematics, an apex graph is a graph that can be made planar by the removal of a single vertex. The deleted vertex is called an apex of the graph. It is an apex, not the apex because an apex graph may have more than one apex; for example, in the minimal nonplanar graphs K_{5} or K_{3,3}, every vertex is an apex. The apex graphs include graphs that are themselves planar, in which case again every vertex is an apex. The null graph is also counted as an apex graph even though it has no vertex to remove.
In topological graph theory, a 1planar graph is a graph that can be drawn in the Euclidean plane in such a way that each edge has at most one crossing point, where it crosses a single additional edge. If a 1planar graph, one of the most natural generalizations of planar graphs, is drawn that way, the drawing is called a 1plane graph or 1planar embedding of the graph.
In graph theory, the thickness of a graph G is the minimum number of planar graphs into which the edges of G can be partitioned. That is, if there exists a collection of k planar graphs, all having the same set of vertices, such that the union of these planar graphs is G, then the thickness of G is at most k. In other words, the thickness of a graph is the minimum number of planar subgraphs whose union equals to graph G.
In graph theory, the Kelmans–Seymour conjecture states that every 5vertexconnected graph that is not planar contains a subdivision of the 5vertex complete graph K_{5}. It is named for Paul Seymour and Alexander Kelmans, who independently described the conjecture; Seymour in 1977 and Kelmans in 1979. A proof has been announced but not yet published.
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