Turán graph

Turán graph
The Turán graph T(13,4)
Named after	Pál Turán
Vertices	$n$
Edges	~$\left(1-{\frac {1}{r}}\right){\frac {n^{2}}{2}}$
Radius	$\left\{{\begin{array}{ll}\infty &r=1\\2&r\leq n/2\\1&{\text{otherwise}}\end{array}}\right.$
Diameter	$\left\{{\begin{array}{ll}\infty &r=1\\1&r=n\\2&{\text{otherwise}}\end{array}}\right.$
Girth	$\left\{{\begin{array}{ll}\infty &r=1\vee (n\leq 3\wedge r\leq 2)\\4&r=2\\3&{\text{otherwise}}\end{array}}\right.$
Chromatic number	$r$
Notation	$T(n,r)$
Table of graphs and parameters

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

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

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

Turán's theorem

Main article: Turán's theorem

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 ${\frac {r\,{-}\,1}{3r}}(2\alpha -1)n^{2}$ edges, if α is sufficiently close to 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.

Special cases

The octahedron, a 3-cross polytope whose edges and vertices form K_2,2,2, 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 K_2n. As Roberts (1969) showed, this graph has boxicity exactly n; it is sometimes known as the Roberts graph. 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 $T(n,\lceil n/3\rceil )$ 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.

Other properties

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.

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.

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. 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.

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.

References

• Chao, C. Y.; Novacky, G. A. (1982). "On maximally saturated graphs". Discrete Mathematics. 41 (2): 139–143. doi:10.1016/0012-365X(82)90200-X.
• Falls, Craig; Powell, Bradford; Snoeyink, Jack (2003). "Computing high-stringency COGs using Turán type graphs" (PDF).
• Keevash, Peter; Sudakov, Benny (2003). "Local density in graphs with forbidden subgraphs" (PDF). Combinatorics, Probability and Computing. 12 (2): 139–153. doi:10.1017/S0963548302005539. S2CID 17854032.
• Moon, J. W.; Moser, L. (1965). "On cliques in graphs". Israel Journal of Mathematics. 3: 23–28. doi:10.1007/BF02760024. S2CID 9855414.
• Nikiforov, Vladimir (2007). "Eigenvalue problems of Nordhaus-Gaddum type". Discrete Mathematics. 307 (6): 774–780. arXiv:math.CO/0506260. doi:10.1016/j.disc.2006.07.035.
• Pór, Attila; Wood, David R. (2005). "No-three-in-line-in-3D". Proc. Int. Symp. Graph Drawing (GD 2004). Lecture Notes in Computer Science no. 3383, Springer-Verlag. pp. 395–402. doi:10.1007/b105810. hdl:11693/27422.
• Roberts, F. S. (1969). Tutte, W.T. (ed.). "On the boxicity and cubicity of a graph". Recent Progress in Combinatorics: 301–310.
• Turán, P. (1941). "Egy gráfelméleti szélsőértékfeladatról (On an extremal problem in graph theory)". Matematikai és Fizikai Lapok. 48: 436–452.
• Witsenhausen, H. S. (1974). "On the maximum of the sum of squared distances under a diameter constraint". American Mathematical Monthly. 81 (10): 1100–1101. doi:10.2307/2319046. JSTOR 2319046.

External links

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