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Graph property

Explore the concept of monotone graph properties, sharp thresholds, and the Erdös-Rényi Graph Model, with a focus on probability transitions in graph properties. Learn how random graphs and thresholds play a crucial role in graph theory.

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Graph property

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  1. Graph property Every Monotone Graph Property has a sharp threshold

  2. Graph property • A graph property is a property of graphs which is closed under isomorphism. • monotone graph property : • Let P be a graph property. • Every graph H on the same set of vertices, which contains G as a sub graph satisfies P as well.

  3. Examples of graph properties • G is connected • G is Hamiltonian • G contains a clique of size t • G is not planar • The clique number of G is larger than that of its complement • the diameter of G is at most s • ... etc .

  4. Erdös – Rényi Graph • Model Erdös - Rényi for random graph • Choose every edge with probability p

  5. Erdös – Rényi Graph • Model Erdös - Rényi for random graph • Choose every edge with probability p

  6. Every Monotone Graph Property has a sharp threshold Ehud Friedgut & Gil Kalai

  7. Definitions • GNP – a graph property • (P) - the probability that a random graph on n vertices with edge probability p satisfies GP. • GG(n,p) - G is a random graph with n vertices and edge probability p.

  8. Main Theorem • Let GNP be any monotone property of graphs on n vertices . If p(GNP) >  then q(GNP) > 1- for q = p + c1log(1/2)/logn absolute constant

  9. Example-Max Clique Probability for choosing an edge • Consider GG(n,p). • The length of the interval of probabilities pfor which the clique number of Gis almost surely k (where k  log n) is of order log-1n. • The threshold interval: The transition between clique numbers k-1 and k. Number of vertices

  10. The probability of having a clique of size k is 1- The probability of having a clique of size k is  • The probability of having a (k + 1)-clique is still small (log-1n). • The value of pmust increase byclog-1n before the probability for having a (k + 1)-clique reaches and another transition interval begins.

  11. Def: Sharp threshold • Sharp threshold in monotone graph property: • The transition from a property being very unlikely to it being very likely is very swift. G satisfies property P G Does not satisfies property P

  12. Conjecture • Let GNP be any monotone property of graphs on n vertices. If p(GNP) >  then q(GNP) > 1- for q = p + clog(1/2)/log2n

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