Turán graph

The Turán graph, denoted by ${\displaystyle T(n,r)}${\displaystyle T(n,r)}, is a complete multipartite graph; it is formed by partitioning a set of ${\displaystyle n}${\displaystyle n} vertices into ${\displaystyle r}${\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 ${\displaystyle q}${\displaystyle q} and ${\displaystyle s}${\displaystyle s} are the quotient and remainder of dividing ${\displaystyle n}${\displaystyle n} by ${\displaystyle r}${\displaystyle r} (so ${\displaystyle n=qr+s}${\displaystyle n=qr+s}), the graph is of the form ${\displaystyle K_{q+1,q+1,\ldots ,q,q}}${\displaystyle K_{q+1,q+1,\ldots ,q,q}}, and the number of edges is

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

For ${\displaystyle r\leq 7}${\displaystyle r\leq 7}, this edge count can be more succinctly stated as ${\displaystyle \left\lfloor \left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}\right\rfloor }${\displaystyle \left\lfloor \left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}\right\rfloor }. The graph has ${\displaystyle s}${\displaystyle s} subsets of size ${\displaystyle q+1}${\displaystyle q+1}, and ${\displaystyle r-s}${\displaystyle r-s} subsets of size ${\displaystyle q}${\displaystyle q}; each vertex has degree ${\displaystyle n-q-1}${\displaystyle n-q-1} or ${\displaystyle n-q}${\displaystyle n-q}. It is a regular graph if ${\displaystyle n}${\displaystyle n} is divisible by ${\displaystyle r}${\displaystyle r} (i.e. when ${\displaystyle s=0}${\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 ${\displaystyle {\frac {r,円{-},1円}{3r}}(2\alpha -1)n^{2}}${\displaystyle {\frac {r,円{-},1円}{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.

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 K_2n. 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) = K_2,2,2 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 ${\displaystyle T(n,\lceil n/3\rceil )}${\displaystyle T(n,\lceil n/3\rceil )} has 3^a2^b 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 R^d, 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.

• Weisstein, Eric W. "Cocktail Party Graph". MathWorld .
• Weisstein, Eric W. "Octahedral Graph". MathWorld .
• Weisstein, Eric W. "Turán Graph". MathWorld .

• Parametric families of graphs
• Extremal graph theory

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