Extremal graph theory

The Turán graph T(n,r) is an example of an extremal graph. It has the maximum possible number of edges for a graph on n vertices without (r+1)-cliques. This is T(13,4).

Extremal graph theory is a branch of the mathematical field of graph theory. Extremal graph theory studies extremal (maximal or minimal) graphs which satisfy a certain property. Extremality can be taken with respect to different graph invariants, such as order, size or girth. More abstractly, it studies how global properties of a graph influence local substructures of the graph.^[1]

Examples

For example, a simple extremal graph theory question is "which acyclic graphs on n vertices have the maximum number of edges?" The extremal graphs for this question are trees on n vertices, which have n − 1 edges.^[2] More generally, a typical question is the following: given a graph property P, an invariant u, and a set of graphs H, we wish to find the minimum value of m such that every graph in H which has u larger than m possess property P. In the example above, H was the set of n-vertex graphs, P was the property of being cyclic, and u was the number of edges in the graph. Thus every graph on n vertices with more than n − 1 edges must contain a cycle.

Several foundational results in extremal graph theory are questions of the above-mentioned form. For instance, the question of how many edges an n-vertex graph can have before it must contain as subgraph a clique of size k is answered by Turán's theorem. Instead of cliques, if the same question is asked for complete multi-partite graphs, the answer is given by the Erdős–Stone theorem.

History

Extremal graph theory, in its strictest sense, is a branch of graph theory developed and loved by Hungarians.

Bollobás (2004)

Extremal graph theory started in 1941 when Turán proved his theorem determining those graphs of order n, not containing the complete graph K_k of order k, and extremal with respect to size (that is, with as many edges as possible).^[3] Another crucial year for the subject was 1975 when Szemerédi proved his result a vital tool in attacking extremal problems.^[3]

Density results

A typical result in extremal graph theory is Turán's theorem. It answers the following question. What is the maximum possible number of edges in an undirected graph G with n vertices which does not contain K₃ (three vertices A, B, C with edges AB, AC, BC; i.e. a triangle) as a subgraph? The complete bipartite graph where the partite sets differ in their size by at most 1, is the only extremal graph with this property. It contains

\left\lfloor {\frac {n^{2}}{4}}\right\rfloor

edges. Similar questions have been studied with various other subgraphs H instead of K₃; for instance, the Zarankiewicz problem concerns the largest graph that does not contain a fixed complete bipartite graph as a subgraph, and the even circuit theorem concerns the largest graph without a fixed-length even cycle. Turán also found the (unique) largest graph not containing K_k which is named after him, namely the Turán graph. This graph is the complete join of "k-1" independent sets (as equi-sized as possible) and has at most

\left\lfloor {\frac {(k-2)n^{2}}{2(k-1)}}\right\rfloor =\left\lfloor \left(1-{\frac {1}{k-1}}\right){\frac {n^{2}}{2}}\right\rfloor

edges. For C₄, the largest graph on n vertices not containing C₄ has

\left({\frac {1}{2}}+o(1)\right)n^{{3/2}}

edges.

Minimum degree conditions

The preceding theorems give conditions for a small object to appear within a (perhaps) very large graph. At the opposite extreme, one might search for conditions which force the existence of a structure which covers every vertex. But it is possible for a graph with

{\binom {n-1}{2}}

edges to have an isolated vertex - even though almost every possible edge is present in the graph - which means that even a graph with very high density may have no interesting structure covering every vertex. Simple edge counting conditions, which give no indication as to how the edges in the graph are distributed, thus often tend to give uninteresting results for very large structures. Instead, we introduce the concept of minimum degree. The minimum degree of a graph G is defined to be

\delta (G)=\min _{{v\in G}}d(v).

Specifying a large minimum degree removes the objection that there may be a few 'pathological' vertices; if the minimum degree of a graph G is 1, for example, then there can be no isolated vertices (even though G may have very few edges).

A classic result is Dirac's theorem, which states that every graph G with n vertices and minimum degree at least n/2 contains a Hamilton cycle.

Notes

References

Bollobás, Béla (2004), Extremal Graph Theory, New York: Dover Publications, ISBN 978-0-486-43596-1 .
Bollobás, Béla (1998), Modern Graph Theory, Berlin, New York: Springer-Verlag, pp. 103–144, ISBN 978-0-387-98491-9 .
Diestel, Reinhard (2010), Graph Theory (4th ed.), Berlin, New York: Springer-Verlag, pp. 169–198, ISBN 978-3-642-14278-9 .
M. Simonovits, Slides from the Chorin summer school lectures, 2006.

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