Wikipedia / Forbidden subgraph problem
Wikipedia / Forbidden subgraph problem
In extremal graph theory, the forbidden subgraph problem-vertex graph can have such that it does not have a subgraph isomorphic is called a forbidden subgraph.
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Definitions
The extremal number is the complete graph is the Turán graph: a complete r-partite vertices, with vertices distributed between parts as equally as possible. The chromatic number such that no two adjacent vertices have the same color.
Upper bounds
Turán's theorem
. Equality cases for Turán's theorem come from the Turán graph.
by considering the chromatic number error.
Erdős–Stone theorem
.
Bipartite graphs
For bipartite graphs. The forbidden subgraph problem for bipartite graphs is known as the Zarankiewicz problem, and it is unsolved in general.
Progress on the Zarankiewicz problem includes following theorem:
Kővári–Sós–Turán theorem..
. This gives the following theorem.
Theorem (Bondy and Simonovits, 1974)..
A powerful lemma in extremal graph theory is dependent random choice. This lemma allows us to handle bipartite graphs with bounded degree in one part:
Theorem (Alon, Krivelevich, and Sudakov, 2003)..
In general, we have the following conjecture.:
Rational Exponents Conjecture (Erdős and Simonovits)..
A survey by Füredi and Simonovits describes progress on the forbidden subgraph problem in more detail.
Lower bounds
There are various techniques used for obtaining the lower bounds.
Probabilistic method
While this method mostly gives weak bounds, the theory of random graphs-free graph.
The probabilistic method. by linearity of expectation-vertex graph exists with at least as many edges as the expected number.
edges in the graph left.
Algebraic constructions
, due to purely geometric reasons, while the graph has a large number of edges to be a strong bound due to way the incidences were defined. The following proof by Erdős, Rényi, and Sős, demonstrates the power of this method.
. Consider the polarity graph isolated vertices, which do not affect the asymptotic value.
.
Theorem (Brown, 1966).
Proof outline..
.
Randomized Algebraic constructions
This technique combines the above two ideas. It uses random polynomial type relations when defining the incidences between vertices, which are in some algebraic set. Using this technique to prove the following theorem.
Theorem.
Proof outline: We take the largest prime power.
-free, and at least one graph exists with the expectation of the number of edges of this resulting graph.
Supersaturation
edges. We introduce Turán density to formalize this notion.
Turán density
is defined to be
is in fact positive and monotone decreasing, so the limit must therefore exist.
error.
Supersaturation Theorem
.
.
Applications
We may solve various forbidden subgraph problems by considering supersaturation-type problems. We restate and give a proof sketch of the Kővári–Sós–Turán theorem below:
Kővári–Sós–Turán theorem..
Proof., then by the Pigeonhole Principle, which is the statement of the theorem.
as well. Other theorems regarding the forbidden subgraph problem which can be solved with supersaturation include:
Generalizations
.
There are also hypergraph. However, the best known upper bound is 0.562, using the technique of flag algebras.