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Turán graph - Wikipedia

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Turán graph

The Turán graph T(13,4)

Named afterPál Turán
Vertices{\displaystyle n}
Edges~{\displaystyle \left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}}
Radius{\displaystyle \left\{{\begin{array}{ll}\infty &r=1\\2&r\leq n/2\\1&{\text{otherwise}}\end{array}}\right.}
Diameter{\displaystyle \left\{{\begin{array}{ll}\infty &r=1\\1&r=n\\2&{\text{otherwise}}\end{array}}\right.}
Girth{\displaystyle \left\{{\begin{array}{ll}\infty &r=1\vee (n\leq 3\wedge r\leq 2)\\4&r=2\\3&{\text{otherwise}}\end{array}}\right.}
Chromatic number{\displaystyle r}
Notation{\displaystyle T(n,r)}
Table of graphs and parameters

The Turán graph, denoted by {\displaystyle T(n,r)}, is a complete multipartite graph; it is formed by partitioning a set of {\displaystyle n} vertices into {\displaystyle r} subsets, with sizes as equal as possible, and then connecting two vertices by an edge if and only if they belong to different subsets. Where {\displaystyle q} and {\displaystyle s} are the quotient and remainder of dividing {\displaystyle n} by {\displaystyle r} (so {\displaystyle n=qr+s}), the graph is of the form {\displaystyle K_{q+1,q+1,\ldots ,q,q}}, and the number of edges is

{\displaystyle \left(1-{\frac {1}{r}}\right){\frac {n^{2}-s^{2}}{2}}+{s \choose 2}}.

For {\displaystyle r\leq 7}, this edge count can be more succinctly stated as {\displaystyle \left\lfloor \left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}\right\rfloor }. The graph has {\displaystyle s} subsets of size {\displaystyle q+1}, and {\displaystyle r-s} subsets of size {\displaystyle q}; each vertex has degree {\displaystyle n-q-1} or {\displaystyle n-q}. It is a regular graph if {\displaystyle n} is divisible by {\displaystyle r} (i.e. when {\displaystyle s=0}).

Turán graphs are named after Pál Turán, who used them to prove Turán's theorem, an important result in extremal graph theory.

By the pigeonhole principle, every set of r + 1 vertices in the Turán graph includes two vertices in the same partition subset; therefore, the Turán graph does not contain a clique of size r + 1. According to Turán's theorem, the Turán graph has the maximum possible number of edges among all (r + 1)-clique-free graphs with n vertices. Keevash & Sudakov (2003) show that the Turán graph is also the only (r + 1)-clique-free graph of order n in which every subset of αn vertices spans at least {\displaystyle {\frac {r\,{-}\,1}{3r}}(2\alpha -1)n^{2}} edges, if α is sufficiently close to 1.[1] The Erdős–Stone theorem extends Turán's theorem by bounding the number of edges in a graph that does not have a fixed Turán graph as a subgraph. Via this theorem, similar bounds in extremal graph theory can be proven for any excluded subgraph, depending on the chromatic number of the subgraph.

The octahedron, a 3-cross polytope whose edges and vertices form K2,2,2, a Turán graph T(6,3). Unconnected vertices are given the same color in this face-centered projection.

Several choices of the parameter r in a Turán graph lead to notable graphs that have been independently studied.

The Turán graph T(2n,n) can be formed by removing a perfect matching from a complete graph K2n. As Roberts (1969) showed, this graph has boxicity exactly n; it is sometimes known as the Roberts graph.[2] This graph is also the 1-skeleton of an n-dimensional cross-polytope; for instance, the graph T(6,3) = K2,2,2 is the octahedral graph, the graph of the regular octahedron. If n couples go to a party, and each person shakes hands with every person except his or her partner, then this graph describes the set of handshakes that take place; for this reason, it is also called the cocktail party graph.

The Turán graph T(n,2) is a complete bipartite graph and, when n is even, a Moore graph. When r is a divisor of n, the Turán graph is symmetric and strongly regular, although some authors consider Turán graphs to be a trivial case of strong regularity and therefore exclude them from the definition of a strongly regular graph.

The class of Turán graphs can have exponentially many maximal cliques, meaning this class does not have few cliques. For example, the Turán graph {\displaystyle T(n,\lceil n/3\rceil )} has 3a2b maximal cliques, where 3a + 2b = n and b ≤ 2; each maximal clique is formed by choosing one vertex from each partition subset. This is the largest number of maximal cliques possible among all n-vertex graphs regardless of the number of edges in the graph; these graphs are sometimes called Moon–Moser graphs.[3]

Every Turán graph is a cograph; that is, it can be formed from individual vertices by a sequence of disjoint union and complement operations. Specifically, such a sequence can begin by forming each of the independent sets of the Turán graph as a disjoint union of isolated vertices. Then, the overall graph is the complement of the disjoint union of the complements of these independent sets.

Chao & Novacky (1982) show that the Turán graphs are chromatically unique: no other graphs have the same chromatic polynomials. Nikiforov (2005) uses Turán graphs to supply a lower bound for the sum of the kth eigenvalues of a graph and its complement.[4]

Falls, Powell & Snoeyink (2003) develop an efficient algorithm for finding clusters of orthologous groups of genes in genome data, by representing the data as a graph and searching for large Turán subgraphs.[5]

Turán graphs also have some interesting properties related to geometric graph theory. Pór & Wood (2005) give a lower bound of Ω((rn)3/4) on the volume of any three-dimensional grid embedding of the Turán graph.[6] Witsenhausen (1974) conjectures that the maximum sum of squared distances, among n points with unit diameter in Rd, is attained for a configuration formed by embedding a Turán graph onto the vertices of a regular simplex.[7]

An n-vertex graph G is a subgraph of a Turán graph T(n,r) if and only if G admits an equitable coloring with r colors. The partition of the Turán graph into independent sets corresponds to the partition of G into color classes. In particular, the Turán graph is the unique maximal n-vertex graph with an r-color equitable coloring.