Extremal graph theory

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) as a subgraph? This graph is known as a Turán graph, it contains

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

edges. Similar questions has been studied with various other subgraphs H instead of K₃. Turán also found the largest graph not containing K_k. This graph has

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

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

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

edges.

References

Bela Bollobas. Extremal Graph Theory. New York: Academic Press, 1978.
Bela Bollobas. Modern Graph Theory. (Chapter 4: Extremal Problems.) New York: Springer, 1998.
Eric W. Weisstein. Extremal Graph Theory. From MathWorld--A Wolfram Web Resource. [1]

Categories: Graph theory

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01-04-2007 01:21:04

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