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Encyclopedia > 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

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

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

edges.

## References

1. Bela Bollobas. Extremal Graph Theory. New York: Academic Press, 1978.
2. Bela Bollobas. Modern Graph Theory. (Chapter 4: Extremal Problems.) New York: Springer, 1998.
3. Eric W. Weisstein. Extremal Graph Theory. From MathWorld--A Wolfram Web Resource. [1] (http://mathworld.wolfram.com/ExtremalGraphTheory.html)

Results from FactBites:

 Graph theory - Wikipedia, the free encyclopedia (1209 words) Informally, a graph is a set of objects called vertices (or nodes) connected by links called edges (or arcs) which can be directed (assigned a direction). Another way to extend basic graphs is by making the edges to the graph directional (A links to B, but B does not necessarily link to A, as in webpages), technically called a directed graph or digraph. Graphs are represented graphically by drawing a dot for every vertex, and drawing an arc between two vertices if they are connected by an edge.
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