POISSON APPROXIMATION FOR THE NUMBER OF INDUCED COPIES OF A FIXED GRAPH IN A RANDOM REGULAR GRAPH

Let Gn,d be a random d-regular graph with n vertices. Given a fixed graph H. W denotes the number of induced copies of H in Gn,d. In this paper, we use Stein-Chen method and Local approach to show that W can approximate by the Poisson distribution and give the bound of this approximation.


Introduction
A random graph G n,p , written by Erdös and Rényi [5] in 1960, is the graph with n labeled vertices 1, 2, .., n and the edges added randomly such that each of the n 2 possible edges exists with probability p, 0 < p < 1.In this work, the author interests in the part of random graph theory in which the degrees of vertices are restricted.Such work focus on regular graphs as the most interesting examples, and the results on regular graphs often extend easily to more general degree sequences.Let 1 ≤ d ≤ n − 1 be two positive integer, a random graph G n,d is obtained by sampling uniformly at random over the set of all simple d-regular graph is on a fixed set of n vertices.We refer the readers to Wormald's survey [3] for more information ( be the set of all possible combinations of v H vertices.For each i ∈ Γ, we define the indicator random variable Then W is the number of copies of H in G n,d .In 2007, Jeong Han Kim, Benny Sodakov and Van Vu [2] proved that the distribution function of W can be approximate by Poisson distribution as the following result. Theorem 1.1 (2).Let H be a strictly balanced graph with v vertices and e ≥ v edges.Let Aut(H) be the number of automorphisms of H and let W be the number of copies of H in a random d-regular graph G n,d .If e → c for some positive constant c, then W converges to P oi λ , the Poisson distribution with mean λ = c e Aut(H) .Clearly, the containing of at least one induced copies of H in G n,d should be closely related to the containing of at least one copies of H in G n,d .
In 2008, Lan XIAO, Guiyin YAN, Yuwen WU and Wei REN [8] showed that the number of induced copies H in random regular graph G n,d can be approximated by Poisson distribution with mean λ = c e aut(H) .The result as following, In 2010 ,Mana Dongoanont and Angkana Boonyued [7] show that the number of copies of H in G n,d can approximate by the Poisson distribution and give the bound of this approximation.The result as following, Theorem 1.3 (7).Let H be a fixed graph with v H vertices and e H ≥ v H edges and let W be the number of copies of In this paper, we use Stein-Chen method and Local approach to show that the number of induced copies H in random regular graph G n,d can approximate by the Poisson distribution and give the bound of this approximation.The following theorem is our main result, where, e F > 0 is the number of edges of any subgraph F ⊆ H such that F isomorphic to H ′ ∩ H ′′ , where H ′ and H ′′ are copies of H sharing at least one edge.
This paper is organized as follows.In section 2, we introduce the Stein-Chen method for Poisson approximation and local approach which use in the proof of main result in section 3.
Throughout this paper, C H stands for an absolute constant depend on H and which possibly different values in different places.

Stein-Chen method and Local Approach
In 1972, Stein [6] gave a wonderful technique to find a bound in the normal approximation to a distribution of a sum of dependent random variables.His technique was depended instead on the elementary differential equation, and in 1975, Chen [4] applied Stein's idea to the Poisson case.
For the rest of the section, unless explicitly stated otherwise, we use the following notation.Γ is the index set, which is finite.In most cases Γ = {1, .., n}, let W = i∈Γ X i where X i are indicator variables and λ = E(W ).
Our goal will be to bound the total variation distance between distribution of W and P oi λ , sup where P oi λ is the Poisson distribution with parameter λ.
Our starting point is the Stein equation for Poisson distribution, which gives, where λ > 0 and j ∈ N ∪ {0}, f A is the unique solution for Stein's equation, A ⊆ N ∪ {0} and I A : N ∪ {0} → R be defined by By substituting j and λ in (1) by any integer-valued random variable W and λ = E(W ), we have From this we get sup To bound the right hand side in (3), a local and coupling approach have been suggested.The first one was used by Chen(1975) and in convenient when the dependence structure of the indicator variables is local (meaning that each indicator is independent of "most" of the other).The idea of combining the Stein's equation with local approach was stated in 2005 by A.D. Barbour and Louis H.Y. Chen [1].It follows that Theorem 2.1 (1).(The local approach) Let W = i∈Γ X i , {X i ; i ∈ Γ} are indicator variables.For each i ∈ Γ, divide Γ \ {i} into two subsets Γ s i and Γ w i , so that, informally, where In next section, we will use Theorem 2.1 to prove our main result.

Proof of Main Result
In this section we prove the our main theorem.Let H be a fixed graph with v H vertices and e H edges such that e H > v H . Let and We divide Γ \ {i} into two subsets as follow and define Lan XIAO, Guiyin YAN, Yuwen WU and Wei REN [8] show that the expectation of the number of induced copies of a graph H in G n,d is asymptotically the same as the expectation of the number of induced copies of H in G(n, p), for all i ∈ Γ and is a number copies of H which spans the vertex i.
By definition of Z i , we have Now, we consider E(X i X j ) for j ∈ Γ s i .

Let H
′ and H ′′ be induced copies of H in G n,d such that spans by the vertices i and j respectively.Case 1.For any two copies H ′ ,H ′′ with at most one vertex in common Case2 : For any two copies H ′ ,H ′′ sharing at least one edge given any subgraph F ⊆ H with e F > 0 such that F isomorphic to H ′ ∩ H ′′ .Then we have Case3 : Given t vertices, there are n 2v−t pairs of edge-disjoint copies H ′ ,H ′′ in K n sharing the t vertices, where t ≥ 2. Hence, E(X i X j ) = P (X i = 1, X j = 1) when d = n δ for δ < 1 such that (1 − δ)(2e H − e F ) ≥ 2v H . Hence: as n → ∞.The proof is complete.

Theorem 1 . 2
(8).Let H be a strictly balanced graph with v vertices and e > v edges.Y H denote the number of induce copies of H in G n,d , and Aut(H) is the number of automorphisms of H.If dn −1+ 1 m(H) → c for some positive constant c, then Y H converges to P oi λ , the Poisson distribution with mean λ = c e Aut(H) .

Theorem 1 . 4 .
Let H be a strictly balanced graph with v vertices and e > v edges.W denotes the number of induced copies of H in G n,d , and Aut(H) is the number of automorphisms of H.If d = n δ where δ < 1 such that (1 − δ)(2e H − e F ) ≥ 2v H , then W converges to Poisson distribution and there exists a constant C H > 0 such that both historical and technical) about this model.As usual, G n,d denotes the random d-regular graph with n labeled vertices and for each vertex has the same degree d.For each H = (V (H), E(H)) be a graph, we use the notation v H = |V (H)| and e H = |E(H)| for the number of vertices and edges, respectively.Call ρ(H) = e H v H the density of H, and m(H) the density of the densest subgraph of H, that is, m(H) = max{ρ(H ′ )|H ′ ⊆ H} and for every H ′ ⊆ H, ρ(H ′ ) ≤ ρ(H), then H is called balanced, and H is called strictly balanced if for every proper subgraph