Extremal graph theory

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Extremal graph theory is a branch of mathematics.

In the narrow sense, extremal graph theory studies the graphs which are extremal among graphs with a certain property. There are various meanings for the word extremal: with the largest number of edges, the largest minimum degree, the smallest diameter, etc. In a broader sense, various other related questions can be included into extremal graph theory. In that case, the term extremal graph theory can encompass a large part of graph theory.

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 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 has 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. Turán also found the (unique) largest graph not containing Kk which is named after him, namely Turán graph. This graph has

\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.

[edit] See also

[edit] References

  1. Béla Bollobás. Extremal Graph Theory. New York: Academic Press, 1978.
  2. Béla Bollobás. Modern Graph Theory, chapter 4: Extremal Problems. New York: Springer, 1998.
  3. Eric W. Weisstein, Extremal Graph Theory at MathWorld.
  4. M. Simonovits, Slides from the Chorin summer school lectures, 2006. [1]