Graph Theory: Threshold Functions and Subgraph Properties, Slides of Design and Analysis of Algorithms

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Let k > 0 be an integer, and let p = p(n) be a function of n such that p ≥ (6k ln n)/n for large n. Then limn→∞P(α ≥ 2 nk ) = 0

For every integer k, there exists a graph H with girth g (H) > k and chromatic number χ(H) > k.

Let P be a graph property- i.e. a class of graphs closed under isomorphism.

Let p = p(n) be a fixed function. If P[G ∈ P] → 1 , as n → ∞, we say that G ∈ P for almost all G ∈ G(n, p).

If P[G ∈ P] → 0 as n → ∞, we say that almost no G ∈ G(n, p) has property P.

For every constant p ∈ (0, 1), and every graph H, almost every G ∈ G(n, p), contains an induced copy of H.

Consider a graph property of the form P = {G : X (G ) ≥ 1 } where X ≥ 0 is a random variable on G (n, p). (Example, connectedness). How can we prove that P has a threshold function t? We study one method here, called second moment method. If we can show that as n → ∞, E (X ) → 0 , then it means, that almost all graphs have property P. (Since P[X ≥ 1] ≤ E (X ), by Markov inequality.) On the other hand we cannot showDocsity.com

The Variance σ^2 of X : σ^2 = E ((X − μ)^2 ). It is a quadratic measure of how much X deviates from its mean.

Chebyshev’s Inequality: For all real λ > 0 , P[|X − μ| ≥ λ] ≤ σ 2 λ^2.

If μ > 0 , for n large, and σ 2 μ^2 →^0 , as^ n^ → ∞, then X (G ) > 0 Since any graph G with X (G ) = 0 satisfies |X (G ) − μ| = μ. So, P[X = 0] ≤ P[|X − μ| ≥ μ] ≤ σ

2 μ^2 →^0 , as^ n^ →^0.

If H is a balanced graph with k vertices, and ≥ 1 edges, then t(n) = n−k/^ is a threshold function for PH.

If k ≥ 3 , then t(n) = n−^1 is a threshold function for the property of containing a k-cycle.

If k ≥ 2 , then t(n) = n^2 /(k−1)^ is a threshold function for the property of containing a Kk.

Let X (G ) denote the number of subgraphs of G isomorphic to H.

Given n ∈ N, let H denote the set of all graphs isomorphic to H whose vertices lie in { 0 , 1 ,... , n − 1 }.

Given H′^ ∈ H, we write H′^ ⊆ G to denote that H′^ itself is a subgraph of G.

The number of isomorphic copies of H on a fixed k set is at most k!.

|H| ≤

(n k

k! ≤ nk^.

Given p = p(n), let γ = p/t, where t = n−k/`. Docsity.com

We have ( kn) nk^ ≥^

1 k!

1 − k− k^1

)k .