UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS WITH CONNECTIONS TO EXTREMAL COMBINATORICS

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1 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS WITH CONNECTIONS TO EXTREMAL COMBINATORICS BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE Abstrat. This paper proves limit theorems for the number of monohromati edges in uniform random olorings of general random graphs. The limit theorems are universal depending solely on the limiting behavior of the ratio of the number of edges in the graph and the number of olors, and works for any graph sequene deterministi or random. The proofs are based on moment alulations whih relates to results of Erdős and Alon on extremal subgraph ounts. As a byprodut of our alulations a simple new proof of a result of Alon, estimating the number of isomorphi opies of a yle of given length in graphs with fixed number of edges, is presented.. Introdution Suppose the verties of a finite graph G = (V, E), with V = n, are olored independently and uniformly at random with olors. The probability that the resulting oloring has no monohromati edge, that is, it is a proper oloring is χ G ()/ n, where χ G () denotes the number of proper olorings of G using -olors. The funtion χ G is the hromati polynomial of G, and is a entral objet in graph theory [5]. A natural generalization of this is to onsider a general oloring distribution p = (p, p 2,..., p ), that is, the probability a vertex is olored with olor a [] is p a independent from the olors of the other verties, where p a 0, and a= p a =. Define P G (p) to be the probability that G is properly olored. P G (p) is related to Stanley s generalized hromati polynomial [30], and under the uniform oloring distribution it is preisely the proportion of proper -olorings of G. Reently, Fadnavis [8] proved that P G (p) is Shur-onvex for every fixed, whenever the graph G is law-free, that is, G has no indued K,3. This implies that for law-free graphs, the probability it is properly olored is maximized under the uniform distribution, that is, p a = / for all a []. The Poisson limit theorems for the number of monohromati subgraphs in a random oloring of a graph sequene G n are appliable when the number of olors grow in an appropriate way ompared to the number of ertain speifi subgraphs in G n. Barbour et al. [5] used Stein s method to show that the number of monohromati edges for the omplete graph onverges to a Poisson random variable. Arratia et al. [4] used Stein s method based on dependeny graphs to prove Poisson approximation theorems for the number of monohromati liques in a uniform oloring of the omplete graph (see also Chatterjee et al. [0]). Poisson limit theorems for the number of general monohromati subgraphs in a random oloring of a graph sequene was studied by Cerquetti and Fortini [9], again using Stein s method. They assumed that the distribution of olors was exhangeable and proved that the number of opies of any partiular monohromati subgraph onverges in distribution to a mixture of Poissons. However, all these results require some tehnial onditions on the subgraph ounts and the oloring probabilities for the error terms in 200 Mathematis Subjet Classifiation. 05C5, 60C05, 60F05, 05D99. Key words and phrases. Combinatorial probability, Extremal ombinatoris, Graph oloring, Limit theorems.

2 2 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE Stein s method to vanish. In partiular, while ounting the number of monohromati edges, the onditions depend on the number of edges and 2-stars, and the oloring probabilities. In this paper we show that these extra onditions are redundant under the uniform oloring sheme; the limiting behavior of the number of monohromati edges is solely governed by the limit of the ratio of the number of edges in the graph and the number of olors... Universal Limit Theorems For Monohromati Edges. Let G n denote the spae of all simple undireted graphs on n verties labeled by [n] := {, 2,, n}. Given a graph G n G n with adjaeny matrix A(G n ) = ((A ij (G n ))) i,j n, denote by V (G n ) the number of verties, and by E(G n ) = i<j n A ij(g n ) the number of edges of G n, respetively. Let p = (p, p 2,..., p ) be a probability vetor, that is, p a 0, and a= p a =. The verties of G n are olored with olors as follows: P(v V (G n ) is olored with olor a {, 2,..., } G n ) = p a, independent from the other verties. The oloring distribution is said to be uniform whenever p a = /, for all a []. If Y i is the olor of vertex i, then N(G n ) := A ij (G n ){Y i = Y j } = {Y i = Y j }, (.) i<j n (i,j) E(G n) denotes the number of monohromati edges in the graph G n. Note that P(N(G n ) = 0) is the probability that G n is properly olored. We study the limiting behavior of N(G n ) as the size of the graph beomes large, allowing the graph itself to be random, under the assumption that the joint distribution of (A(G n ), Y n ) is mutually independent, where Y n = (Y, Y 2,..., Y n ) are i.i.d. random variables with P(Y = a) = p a, for all a []. Note that this setup inludes the ase where {G, G 2,...} is a deterministi (non-random) graph sequene, as well. An appliation of the easily available version of Stein s method, similar to that in Cerquetti and Fortini [9], gives a general limit theorem for N(G n ) that works for all olor distributions (Theorem 2.). However, this result, like all other similar results in the literature, requires several onditions involving the number of edges and 2-stars in the graph G n, even when the oloring sheme is uniform. One of the main ontribution of this paper is in showing that these extra onditions are, in fat, redundant under the uniform oloring sheme, and the limiting behavior is solely governed by the limit of E(G n ) /. Theorem.. Let G n G n be a random graph sampled aording to some probability distribution over G n. Then under the uniform oloring distribution, that is, p a = /, for all a [], the following is true: 0 if N(G n ) D E(G n) P 0, if E(G n) P, W if E(G n) D Z; where P(W = k) = k! E(e Z Z k ). In other words, W is distributed as a mixture of Poisson random variables mixed over the random variable Z. Theorem. is universal beause it only depends on the limiting behavior of E(G n ) / and it works for any graph sequene {G n } n, deterministi or random. The theorem is proved using the method of moments, that is, the onditional moments of N(G n ) are ompared with onditional moments of the random variable M(G n ) := i<j n A ij(g n )Z ij, where {Z ij } (i,j) E(Gn) are independent Ber(/). The ombinatorial quantity to bound during the moment alulations is the number of isomorphi opies of a graph H in another graph G, to be denoted by N(G, H). Using

3 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS 3 properties of the adjaeny matrix of G we estimate N(G, H), when H = C g is a g-yle, This result is then used to show the asymptoti loseness of the onditional moments of N(G n ) and M(G n ). Theorem. asserts that if E(G n) P, then N(G n ) goes to infinity in probability. Sine a Poisson random variable with mean growing to infinity onverges to a standard normal distribution after entering and saling by the mean and the standard deviation, it is natural to wonder whether the same is true for N(G n ). This is not true in general if E(G n ) / goes to infinity, with fixed. Proposition 6. shows that the limiting distribution of the number of monohromati edges in the omplete graph properly saled is asymptotially a hi-square with ( ) degrees of freedom, when is fixed. On the other hand, if, Stein s method for normal approximation an be used to prove the asymptoti normality of N(G n ). However, as before, in the off-the-shelf version of Stein s method an extra ondition is needed on the struture of the graph, even under the uniform oloring sheme. Nevertheless, as in the Poisson limit theorem, the normality of the standardized random variable N(G n ) is universal and an also be proved by a method of moments argument. Theorem.2. Let G n G n be a random graph sampled aording to some probability distribution over G n, and N(G n ) as defined before. Then for any uniform -oloring of G n, with and E(G n ) / P, Z n := ( N(G n ) E(G ) n) D N(0, ). E(Gn ) / In the proof of Theorem.2 the onditional entral moments of N(G n ) are ompared with the onditional entral moments of M(G n ). In this ase a ombinatorial quantity involving the number of multi-subgraphs of G n show up. Bounding this quantity requires extensions of Alon s [2, 3] results to multi-graphs and leads to results in graph theory whih may be of independent interest..2. Connetions to Extremal Combinatoris. The ombinatorial quantity that shows up in the moment omputations for the Poisson limit theorem is N(G, H), the number of isomorphi opies of a graph H in another graph G. The quantity N(l, H) := sup G: E(G) =l N(G, H) is a well-known objet in extremal graph theory that was first studied by Erdős [7] and later by Alon [2, 3]. Alon [2] showed that for any simple graph H there exists a graph parameter γ(h) suh that N(l, H) = Θ(l γ(h) ). Friedgut and Kahn [9] extended this result to hypergraphs and identified the exponent γ(h) as the frational stable number of the hypergraph H. Alon s result an be used to obtain a slightly more diret proof of Theorem.. However, our estimates of N(G, C g ) using the spetral properties of G lead to a new and elementary proof of the following result of Alon [2]: Theorem.3 (Theorem B, Alon [2]). If H has a spanning subgraph whih is a disjoint union of yles and isolated edges, then N(l, H) = ( + O(l /2 )) where Aut(H) denotes the number of automorphisms of H. Aut(H) (2l) V (H) /2, The above theorem alulates the exat asymptoti behavior of N(l, H) for graphs H whih have a spanning subgraph onsisting of a disjoint union of yles and isolated edges. There are only a handful of graphs for whih suh exat asymptotis are known [2, 3]. Alon s proof in [2] uses a series of ombinatorial lemmas. We hope the short new proof presented in this paper is of independent interest.

4 4 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE The quantity γ(h) is a well studied objet in graph theory and disrete optimization and is related to the frational stable set polytope [28]. While proving Theorems. and.2 we disover several new fats about the exponent γ(h), whih might be of independent interest in graph theory. Alon [3] showed that γ(h) E(H), and the equality holds if and only if H is a disjoint union of stars. This is improved to γ(h) V (H) ν(h), where ν(h) is the number of onneted omponents of H and the ondition for equality remains the same. This is proved in Corollary 5.2 and used later to give an alternative proof of Theorem.. In fat, the universality of the Poisson limit neessitates γ(h) < V (H) ν(h) for all graphs with a yle. In a similar manner, the universal normal limit leads to the following interesting disovery about γ(h). Suppose H has no isolated verties: if γ(h) > 2 E(H), then H has a vertex of degree (Lemma 6.2). This result is true for simple graphs as well as multi graphs (with a similar definition of γ for multi-graphs). This result is sharp, in the sense that there are simple graphs with no leaves suh that γ(h) = E(H) /2. That graphs with γ(h) > E(H) /2 always have a leaf is a surprising and fortunate phenomenon, and is exatly what is needed for the proof of universal normality..3. Other Monohromati Subgraphs. Theorems. and.2 determine the universal asymptoti behavior of the number of monohromati edges under independent and uniform oloring of the verties. However, the situation while ounting opies of other monohromati subgraphs is quite different. Even under uniform oloring, the limit need not be Poisson mixture. This is illustrated in the following proposition where we show that the number of monohromati r-stars in a uniform oloring of an n-star onverges to a polynomial in Poissons, whih is not a Poisson mixture. Proposition.4. Let G n = K,n, with verties labeled by [n]. Under the uniform oloring sheme, the random variable T r,n whih ounts the number of monohromati r-stars in G n satisfies: n 0 if 0, D n T r,n if, X(X ) (X r+) n r! if λ, where X P oisson(λ). Examples with other monohromati subgraphs are also onsidered and several interesting observations are reported. We onstrut a graph G n where the number of monohromati g-yles (g 3) in a uniform -oloring of G n onverges in distribution to a non-trivial mixture of Poisson even when N(G n, C g ) / g onverges to a fixed number λ. This is in ontrast to the situation for edges, where the number of monohromati edges onverges to P oisson(λ) whenever E(G n ) / λ. We believe that a Poisson-mixture universality holds for yles as well, that is, the number of monohromati g-yles in a uniform random oloring of any graph sequene G n onverges in distribution to a random variable whih is mixture of Poisson, whenever N(G n, C g ) / g λ > Organization of the Paper. The paper is organized as follows: Setion 2 disusses Stein s method approah for studying the limiting behavior of N(G n ). Setion 3, proves Theorem. and Setion 4 illustrates it for various ensembles of fixed and random graphs. Setion 5 disusses the onnetions with the subgraph ounting problem from extremal ombinatoris and gives a new proof of Theorem.3. The proof of Theorem.2, where the universal normality of N(G n ) is established, is in Setion 6. Finally, Setion 7 proves Proposition.4, onsiders other examples about ounting monohromati yles, and disusses possible diretions for future researh. An appendix gives details on onditional and unonditional onvergene of random variables.

5 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS 5 2. General Poisson Approximation Using Stein s Method Cerquetti and Fortini [9] proved a Poisson limit theorem for the number of monohromati subgraphs in a random oloring of a graph sequene using Stein s method, under the assumption that the distribution of the olors is exhangeable. A similar appliation of Stein s method gives a general Poisson limit theorem for N(G n ) as well. Theorem 2.. Let G n G n be a random graph sampled aording to some distribution, and Z be a non-negative random variable. If T (G n ) denotes the set of 2-stars in G n, then the following is true if p 2 E(G n ) P 0, P(N(G n ) = 0) 0 if p 2 E(G n ) P, ( p 3 + p 2 2 ) T (G n) = o P ( p 2 2 E(G n) ), E(e Z ) if p 2 0, p 2 E(G n ) D Z, ( p 3 + p 2 2 ) T (G n) P 0, where p r = k= pr k, with r N. More generally, in the third ase N(G n) D W, where W is a non negative integer valued random variable with P(W = k) = k! E(e Z Z k ), that is, W is distributed as a Poisson random variable with parameter Z. Proof. Note that P(N(G n ) > 0 G n ) E(N(G n ) G n ) = p 2 E(G n ), from whih the first ase follows. For the seond part, note that E(N(G n ) 2 G n ) = {Y i = Y j }{Y k = Y l } e =(i,j),e 2 =(k,l) = p 2 E(G n ) + p 3 T (G n ) + p 2 2( E(G n ) 2 T (G n ) E(G n ) ) = ( p 2 p 2 2) E(G n ) + ( p 3 p 2 2) T (G n ) + p 2 2 E(G n ) 2 p 2 E(G n ) + ( p 3 + p 2 2) T (G n ) + p 2 2 E(G n ) 2 The result then follows from the inequality P(N(G n ) > 0 G) E(N(Gn) Gn)2 E(N(G n) 2 G n) and the given onditions. For the third ase, using [0, Theorem 5] with Z n := E(G n ) p 2 we have for all k N, P(N(G n) = k G n ) k! e Zn Zn k 2( p 3 + p 2 2) T (G n ) + 2 p 2 2 E(G n ). The right hand side onverges to 0 in probability from given onditions, and by the bounded onvergene theorem the onlusion follows. For the uniform oloring sheme, the above theorem takes the following form. It establishes a Poisson limit theorem with onditions that depend on the edges, 2-stars, and the number of olors. Corollary 2.2. (Uniform Coloring) Let G G n be a random graph sampled aording to some distribution, and Z be a non-negative random variable. Then under the uniform oloring distribution, that is, p a = /, for all a [], the following holds P(N(G n ) = 0) if E(G n) P 0, 0 if E(G n) P E(e Z ) if E(G n) D Z, 2 T (G n ) P 0.

6 6 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE Remark 2.. The ondition E(G n) D Z ensures that the number of edges grows as the number of verties inreases, and is satisfied by almost all fixed and random graph ensembles in pratie. The ondition T (G 2 n ) onverges in probability to 0, is a tehnial ondition arising from Stein s method that is required for the proof. For an expliit example where the onlusion holds even though assumptions do not, let G n = K,n be the star graph with verties labelled by [n] and the entral vertex labelled by. Let = n, and p a = /n, for a []. Then E(G n) = /n, and T (G 2 n ) = (n )(n 2)/n 2. Thus, in this ase the last ondition of Corollary 2.2 does not hold. If Y i denotes the olor of vertex i [n], then N(G n ) := n i=2 {Y = Y i }. Observe that all the summands are independent given Y, and so for any s (0, ) Es N(Gn) = E n E(s {Y =Y j } Y ) = j=2 ( n + s n) n e s = k=0 k!e sk. Therefore, N(G n ) onverges to a Poisson distribution with parameter, whih is not predited by Corollary 2.2. On the other hand, Theorem., whih is proved in the next setion, overs this ase and all other ases where the Poisson limit theorem for monohromati edges holds. 3. Universal Poisson Approximation For Uniform Colorings: Proof of Theorem. This setion determines the limiting behavior of P(N(G n ) = 0) under minimal onditions. Using the method of moments, we show that N(G n ) has a universal threshold whih depends only on the limiting behavior of E(G n ) /, and a Poisson limit theorem holds at the threshold. Let G n G n be a random graph sampled aording to some probability distribution. For every n N fixed, for (i, j) E(G n ) define the olletion of random variables {Z ij, (i, j) E(G n )}, where Z ij are i.i.d. Ber(/), and set M(G n ) := A ij (G n )Z ij. i<j n (3.) The proof of Theorem. is given in two parts: The first part ompares the onditional moments of N(G n ) and M(G n ), given the graphs G n, showing that they are asymptotially lose, when E(G n ) / D Z. The seond part uses this result to omplete the proof of Theorem. using some tehnial properties of onditional onvergene (see Lemma 8.). 3.. Computing and Comparing Moments. This setion is devoted to the omputation of onditional moments of N(G n ) and M(G n ), and the omparison of these. To this end, define for any fixed number k, A k B as A C(k)B, where C(k) is a onstant that depends only on k. Let G n G n be a random graph sampled aording to some probability distribution. For any fixed subgraph H of G n, let N(G n, H) be the number of isomorphi opies of H in G n, that is, N(G n, H) := {G n [S] = H}, S E(G n): S = E(H) where the sum is over subsets S of E(G n ) with S = E(H), and G n [S] is the subgraph of G n indued by the edges of S. Lemma 3.. Let G n G n be a random graph sampled aording to some probability distribution. For any k, let H k to be the olletion of all graphs with at most k edges and no isolated verties. Then E(N(G n ) k G n ) E(M(G n ) k G n ) k N(G n, H), (3.2) V (H) ν(h) H H k, H has a yle

7 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS 7 where ν(h) is the number of onneted omponents of H. Proof. Using the multinomial expansion and the definition of H k, ( k ) E(N(G n ) k G n ) = E {Y ir = Y jr } G n Similarly, E(M(G n ) k G n ) = (i,j ) E(G n) (i 2,j 2 ) E(G n) (i,j ) E(G n) (i 2,j 2 ) E(G n) (i k,j k ) E(G n) (i k,j k ) E(G n) r= (3.3) ( k ) E Z irjr, (3.4) If H is the simple subgraph of G n indued by the edges (i, j ), (i 2, j 2 ),... (i k, j k ). Then ( k ) ( k ) E {Y ir = Y jr } G n = and E Z V (H) ν(h) irjr = E(H). r= The result now follows by taking the differene of (3.3) and (3.4), and noting that in any graph H, E(H) V (H) ν(h) and equality holds if and only if H is a forest. Remark 3.. Observe that if G n is a forest (disjoint union of trees), then by the above lemma, E(M(G n ) k G n ) = E(N(G n ) k G n ), and onsequently the laws of M(G n ) and N(G n ) are exatly the same when G n is a forest. In partiular, this means that for a forest P(N(G n ) = 0 G n ) = ( r= ) E(Gn), for every, n. Note that under the uniform oloring sheme P(N(G n ) = 0 G n ) = n χ(g n, ), where χ(g n, ) is the number of proper -oloring of the graph G n. Therefore, determining exat or asymptoti expressions of P(N(G n ) = 0 G n ), when is a fixed onstant, for a graph G amounts to ounting the number of proper -olorings of G. Though this is easy for trees, in general it is a notoriously hard problem, and is known to be #P-omplete (refer to the survey by Frieze and Vigoda [20] and the referenes therein for the various hardness results and approximate ounting tehniques related to proper graph olorings). Lemma 3. shows that bounding the differene of the onditional moments of N(G n ) and M(G n ) entails bounding N(G n, H), the number of opies of a subgraph H in G n. The next lemma estimates the number of opies of a yle C g in G n. Lemma 3.2. For g 3 and G n G n let N(G n, C g ) be the number of g-yles in G n. Then N(G n, C g ) 2g (2 E(G n) ) g/2. Proof. Let A := A(G n ) be the adjaeny matrix of G n. Note that n i= λ2 i = tr(a2 ) = 2 E(G n ), where λ λ n are the ordered absolute eigenvalues of A. Note that tr(a g ) ounts the number of walks of length g in G n, and so eah yle in G n is ounted 2g times. Thus, as an upper bound, N(G n, C g ) 2g tr(ag ) = 2g n λ g i n 2g sup λ i g 2 where the last step uses sup i λ i g 2 = (sup i λ i 2 ) g/2 (2 E(G n ) ) g/2. i= i i= r= λ 2 i 2g (2 E(G n) ) g/2,

8 8 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE Remark 3.2. In extremal graph theory, the study of N(G, H), for arbitrary graphs G and H, was initiated by Erdős [7], and later arried forward by Alon [2, 3]. In fat, Lemma 3.2 is a speial ase of Theorem B of Alon [2]. In Setion 5 the proof of Lemma 3.2 is used to give a new and short proof of Theorem B. For a given simple graph H, the notation A H B will mean A C(H) B, where C(H) is a onstant that depends only on H. Lemma 3.2 gives a bound on N(G n, H) in terms of E(G n ) for arbitrary subgraphs H of G n. Lemma 3.3. For any fixed onneted subgraph H, let N(H, G n ) be the set of opies of H in G n. Then N(G n, H) H E(G n ) V (H). (3.5) Furthermore, if H has a yle of length g 3, then N(G n, H) H E(G n ) V (H) g/2. (3.6) Proof. The first bound on N(H, G n ) an be obtained by a rude ounting argument as follows: First hoose an edge of G n in E(G n ) whih fixes 2 verties of H. Then the remaining V (H) 2 verties are hosen arbitrarily from the set of allowed V (G n ) verties, giving the bound (H) ( V N(G n, H) 2 2 ) E(Gn ) ( ) V (Gn ) V (H) 2 (H) ( V 2 2 ) E(Gn ) ( ) 2 E(Gn ) H E(G n ) V (H), V (H) 2 (H) ( V where we have used the fat that the number of graphs on V (H) verties is at most 2 2 ). Next, suppose that H has a yle of length g 3. Choosing a yle of length g arbitrarily from G n, there are V (G n ) verties from whih the remaining V (H) g verties are hosen arbitrarily. Sine the edges among these verties are also hosen arbitrarily, the following rude upper bound holds ( ) (H) ( V N(G n, H) 2 2 ) g 2 E(Gn ) N(G n, C g ) V (H) g where the last step uses Lemma 3.2. H V (H) g N(G n, C g ) E(G n ) H E(G n ) V (H) g/2. (3.7) The above lemmas, imply the most entral result of this setion: the onditional moments of M(G n ) and N(G n ) are asymptotially lose, whenever E(G n ) / D Z. Lemma 3.4. Let M(G n ) and N(G n ) be as defined before, with E(G n ) / D Z, then for every fixed k, E(N(G n ) k G n ) E(M(G n ) k G n ) P 0. Proof. By Lemma 3. E(N(G n ) k G n ) E(M(G n ) k G n ) H H k, H has a yle N(G n, H) V (H) ν(h), where ν(h) is the number of onneted omponents of H. As the sum over H H k is a finite sum, it suffies to show that for a given H H k with a yle N(G n, H) = o P ( V (H) ν(h) ).

9 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS 9 To this end, fix H H k and let H, H 2,..., H ν(h) be the onneted omponents of H. Without loss of generality, suppose the girth of G, g(g) = g(h ) = g 3. Lemma 3.3 then implies that N(G n, H) ν(h) i= N(H i, G n ) H E(G n ) V (H ) g/2 ν(h) i=2 E(G n ) V (H i) H E(G n ) V (H) ν(h)+ g/2 (3.8) whih is o p ( V (H) ν(h) ) sine g/2 > 0. Remark 3.3. The above proof shows that the error rate between the differene of onditional moments is better when g is larger, that is, the Poisson approximation is more aurate on graphs with large girth. In his 98 paper, Alon [2] proved that for every fixed H, N(G n, H) 2 (H) E(G n ) γ(h), where 2 (H) and γ(h) are onstants depending only on the graph H. Setion 5 below shows that by plugging in this estimate and using the property of γ(h), it is possible to obtain a diret proof 3.4 that obviates the alulations in Lemma 3.2 and Lemma 3.3. This gives slightly better error rates between the differene of the onditional moments Completing the Proof of Theorem.. The results from the previous setion are used here to omplete the proof of Theorem.. The three different regimes of E(G n ) / are treated separately as follows: E(G n) P 0. The result follows diretly from Corollary E(G n) P. It follows from the proof of Theorem 2. that E(N 2 n G n )/E(N n G n ) 2. This implies that N n /E(N n G n ) onverges in probability to, and so N n onverges to in probability, as E(N n G n ) = E(G n) P E(G n) D Z, where Z is some random variable. In this regime the limiting distribution of N(G n ) is a mixture of Poisson. As the Poisson distribution an be uniquely identified by moments, from Lemma 3.4 it follows that onditional on { E(G n ) / λ}, N(G n ) onverges to P oisson(λ) for every λ > 0. However, this does not immediately imply the unonditional onvergene of N(G n ) to a mixture of Poisson. In fat, a tehnial result, detailed in Lemma 8., and onvergene of M(G n ) to a Poisson mixture is neessary to omplete the proof. To begin with, reall that a random variable X is a mixture of Poisson with mean Z, to be denoted as P oisson(z), if there exists a non-negative random variable Z suh that P(X = k) = E ( k! e Z Z k The following lemma shows that M(G n ) onverges to P oisson(z) and satisfies the tehnial ondition needed to apply Lemma 8.. Lemma 3.5. Let M(G n ) be as defined in (3.) and E(G n) D Z. Then M(G n ) D P oisson(z), and further for any ɛ > 0, t R, ( t k ) lim sup lim sup P k n k! E(M(G n) k G n ) > ɛ = 0. ).

10 0 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE ( ) E(Gn) Proof. For any t R, Ee itm(gn) = EE(e itm(gn) G n ) = E + eit = EZn, where Z n := ( ) E(Gn) + eit satisfies Zn. Sine ) ( eit e log Z n = E(G n ) log ( it ( )) D + = E(G n ) + O 2 (e it )Z, by the dominated onvergene theorem Ee itm(gn) = EZ n Ee (eit )Z, whih an be easily heked to be the generating funtion of a random variable with distribution P oisson(z). Thus, it follows that M(G n ) D P oisson(z). Proeeding to hek the seond onlusion, reall the standard identity z k = k j=0 S(k, j)(z) j, where S(, ) are Stirling numbers of the seond kind and (z) j = z(z ) (z j + ). In the above identity, setting z = M(G n ), taking expetation on both sides onditional on G n, and using the formula for the Binomial fatorial moments, k E(M(G n ) k G n ) = S(k, j)( E(G n ) ) j j. j=0 The right hand side onverges weakly to k j=0 S(k, j)zj. This is the k-th mean of a Poisson random variable with parameter Z. Using the formula for the Poisson moment generating funtion, for any Z 0 and any t R we have k=0 t k k! k j=0 S(k, j)z j = e Z(et ) < = tk k! as k. Thus, applying Fatou s Lemma twie gives ( t k ) ( lim sup lim sup P k n k! E(M(G n) k t k G n ) > ɛ lim sup P k k! and so the proof of the lemma is omplete. k j=0 S(k, j)z j a.s. 0, ) k S(k, r)z r > ɛ = 0, Now, take U n,k = E(N(G n ) k G n ) and V n,k = E(M(G n ) k G n ), and observe that (8.) and (8.2) hold by Lemma 3.4 and Lemma 3.5, respetively. As M(G n ) onverges to P oisson(z), this implies that N(G n ) onverges to P oisson(z), and the proof of Theorem. is ompleted. Remark 3.4. Theorem. shows that the limiting distribution of the number of monohromati edges onverges to a Poisson mixture. In fat, Poisson mixtures arise quite naturally in several ontexts. It is known that the Negative Binomial distribution is distributed as P oisson(z), where Z is a Gamma random variable with integer values for the shape parameter. Greenwood and Yule [22] showed that ertain empirial distributions of aidents are well-approximated by a Poisson mixture. Le-Cam and Traxler [25] proved asymptoti properties of random variables distributed as mixture of Poisson. Poisson mixtures are widely used in modeling ount panel data (refer to the reent paper of Burda et al. [7] and the referenes therein), and have appeared in other applied problems as well []. r=0 4. Examples: Appliations of Theorem. In this setion we apply Theorem. to different deterministi and random graph models, and determine the speifi nature of the limiting Poisson distribution.

11 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS Example. (Birthday Problem) When the underlying graph G is the omplete graph K n on n verties, the above oloring problem redues to the well-known birthday problem. By replaing the olors by birthdays, eah ourring with probability /, the birthday problem an be seen as oloring the verties of a omplete graph independently with olors. The event that two people share the same birthday is the event of having a monohromati edge in the olored graph. In birthday terms, P(N(K n ) = 0) is preisely the probability that no two people have the same birthday. Theorem. says that under the uniform oloring for the omplete graph P(N(K n ) = 0) e n2 /2. Therefore, the maximum n for whih P(N(K n ) = 0) /2 is approximately 23, whenever = 365. This reonstruts the lassial birthday problem whih an also be easily proved by elementary alulations. For a detailed disussion on the birthday problem and its various generalizations and appliations refer to [, 5, 2, 3, 4] and the referenes therein. Example 2. (Birthday Coinidenes in the US Population) Consider the following question: What is the hane that there are two people in the United States who (a) know eah other, (b) have the same birthday, () their fathers have the same birthday, (d) their grandfathers have the same birthday, and (e) their great grandfathers have the same birthdays. We will argue that this seemingly impossible oinidene atually happens with almost absolute ertainty. The population of the US is about n=400 million and it is laimed that a typial person knows about 600 people [2, 23]. If the network G n of who knows who is modeled as an Erdős-Renyi graph, this gives p = and E( E(G n ) ) = =.2 0. The 4-fold birthday oinidene amounts to = (365) 4 olors and Poisson approximation has λ = E( E(G n ) )/ So the hane of a math is approximately e λ 99.8%, whih means that almost surely there are two friends in the US who have this sensational 4-fold birthday mathes. Going bak one more generation, we now alulate the probability that there are two friends who have a 5-fold birthday oinidene between their respetive anestors. This amounts to = (365) 5 and Poisson approximation shows that the hane of a math is approximately e λ.8%. This implies that even a miraulous 5-fold oinidene of birthdays is atually likely to happen among the people of the US. Example 3. (Random Regular Graphs) When G n onsists of the set all d-regular graphs on n verties and sampling is uniformly on this spae, then under the uniform oloring distribution with, Theorem. gives nd if 0, nd P(N(G n ) = 0) 0 if, e b/2 nd if b. Example 4. (Sparse Inhomogeneous Random Graphs) A general model for sparse random graphs is the following: every edge (i, j) is present independently with probability n f( i n, j n ), for some symmetri ontinuous funtion f : [0, ] 2 [0, ] (see Bollobas et al. [6]). Under the uniform oloring distribution, n E(G n) 2 Consequently, theorem. gives P(N(G n ) = 0) ˆ ˆ 0 e (b/2) 0 0 f(x, y)dxdy. n if 0, n 0 if, 0 f(x,y)dxdy n if b.

12 2 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE Note that this model inludes as a speial ase the Erdős-Renyi random graphs G(n, λ/n), by taking the funtion f(x, y) = λ. Example 5. (Dense graph limits) Limits of dense graphs was developed reently by Lovász and o-authors [26], where a random graph sequene G n onverges in ut-metri to a random symmetri measurable funtion W : [0, ] 2 [0, ]. Then n 2 E(G n) 2 ˆ ˆ 0 0 W (x, y)dxdy. Thus whenever, by Theorem. we have: n if 2 0, P(N(G n ) = 0) n 0 if 2, E(e (b/2) 0 0 W (x,y)dxdy n ) if 2 b. Thus the result holds irrespetive of the speifi model on random graphs, as long as it onverges in the sense of the ut metri. In partiular, the result implies diretly in the following examples: Inhomogenous random graphs: Let f : [0, ] 2 [0, ] be a symmetri ontinuous funtion. Consider the random graph model where and edge (i, j) is present with probability f( i n, j n ) and the uniform oloring distribution. Therefore, whenever, by Theorem. we have: n if 2 0, P(N(G n ) = 0) n 0 if 2, e (b/2) 0 0 f(x,y)dxdy n if 2 b. Note that this model inludes as a speial ase the Erdős-Renyi random graphs G(n, p), by taking the funtion f(x, y) = p. Graph Limits: Let {U i } n i.i.d. i= U(0, ), and let f be a symmetri ontinuous funtion, and onsider the random graph model where given U the edges are mutually independent, with an edge (i, j) being present with probability f(u i, U j ). These random graphs also onverge to f in probability with respet to the ut metri, and onsequently the same onlusions as in the inhomogeneous random graph model hold in this ase. Refer to Lovász s reent book [26] for a omplete desription of the theory of graph limits. Example 6. (Galton-Watson Trees) Let G n be a Galton-Watson tree trunated at height n, and let ξ denote a generi random variable from the off-spring distribution. Assume further that µ := Eξ >. This ensures that the total progeny up to time n (whih is also the number of edges in G n ) grows with n. Letting {Z i } i=0 denote the size of the i-th generation, the total progeny up to time n is Y n := n i=0 Z i. Assuming that the population starts with one off-spring at time 0, that is, Z 0, Z n /µ n is a non-negative martingale ([6, Lemma 4.3.6]). It onverges almost surely to a finite valued random variable Z, by [6, Theorem 4.2.9], whih readily implies Y n /µ n+ onverges almost surely to Z /(µ ). Thus, Theorem. gives µ if n 0, µ P(N(G n ) = 0) 0 if n, Ee bµ µ Z µ if n b. Note in passing that Z 0 if and only if E(ξ log ξ) = ([6, Theorem 4.3.0]). Thus, to get a nontrivial limit the neessary and suffiient ondition is E(ξ log ξ) <.

13 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS 3 5. Connetions to Extremal Graph Theory In the method of moment alulations of Lemma 3., we enounter the quantity N(G, H), the number of isomorphi opies of H in G. More formally, given two graphs G = (V (G), E(G)) and H = (V (H), E(H)), N(G, H) := {G[S] = H}, S E(G): S = E(H) where the sum is over subsets S of E(G) with S = E(H), and G[S] is the subgraph of G indued by the edges of S. For a positive integer l E(H), define N(l, H) := sup G: E(G) =l N(G, H). For the omplete graph K h, Erdős [7] determined N(l, K h ), whih is also a speial ase of the Kruskal-Katona theorem, and posed the problem of estimating N(l, H) for other graphs H. This was addressed by Alon [2] in 98 in his first published paper. Alon studied the asymptoti behavior of N(l, H) for fixed H, as l tends to infinity. He identified the orret order of N(l, H), for every fixed H, by proving that: Theorem 5. (Alon [2]). For a fixed graph H, there exists two positive onstants = (H) and 2 = 2 (H) suh that for all l E(H), l γ(h) N(l, H) 2 l γ(h), (5.) where γ(h) = 2 ( V (H) + δ(h)), and δ(h) = max{ S N H(S) : S V (H)}. Friedgut and Kahn [9] extended this result to hypergraphs, and identified the orret exponent γ(h) as the frational stable number of the hypergraph H. Using the above theorem or the definition of γ(h) it is easy to show that γ(h) E(H), and the equality holds if and only if H is a disjoint union of stars (Theorem, Alon [3]). The following orollary gives a sharpening of Theorem of [3]: Corollary 5.2. For every graph H, γ(h) V (H) ν(h), where ν(h) is the number of onneted omponents of H. Moreover, the equality holds if and only if H is a disjoint union of stars. Proof. Suppose H, H 2,..., H ν(h) are the onneted omponents of H. Fix i {, 2,..., ν(h)}. Sine H i is onneted, for every S V (H i ), S N Hi (S) V (H i ) 2. This implies that δ(h) = ν(h) i= δ(h i) V (H) 2ν(H), and γ(h) V (H) ν(h). Now, if H is a disjoint union of stars with ν(h) onneted omponents, then by Theorem of Alon [3], γ(h) = E(H) = V (H) ν(h). Conversely, suppose that γ(h) = V (H) ν(h). If H has a yle of length g 3, then from (3.8) and Theorem 5. γ(h) V (H) ν(h) + g/2 < V (H) ν(h). Therefore, H has no yle, that is, it is a disjoint union of trees. This implies that γ(h) = V (H) ν(h) = E(H), and from Theorem of Alon [3], H is a disjoint union of stars. 5.. Another Proof of Theorem. Without Lemma 3.. Theorem 5. and Corollary 5.2 give a diret proof of Lemma 3.4, whih does not require the subgraph ounting Lemma 3.. With M(G n ) and N(G n ) be as defined before, and E(G n ) / D Z, for every fixed k, E(N(G n ) k G n ) E(M(G n ) k G n ) k N(G n, H) V (H) ν(h) k H H k, H has a yle H H k, H has a yle E(G n) γ(h), V (H) ν(h)

14 4 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE where the last inequality follows from Theorem 5.. As the sum is over all graphs H whih are not a forest, it follows from Corollary 5.2 that γ(h) < V (H) ν(h). Therefore, every term in the sum goes to zero as n, and, sine H H k is a finite sum, Lemma 3.4 follows A New Proof of Theorem.3 Using Lemma 3.. This setion gives a short proof of Theorem.3 using Lemma 3.. The proof uses spetral tehniques and is quite different from the proof in Alon [2] Proof of Theorem.3. Let F be the spanning subgraph H, and let F, F 2,... F q, be the onneted omponents of F, where eah F i is a yle or an isolated edge, for i {, 2,..., q}. Consider the following two ases: Case : F i is an isolated edge. Then for any graph G with E(G) = l, N(G, F i ) = l = Case 2: F i is a yle of length g 3. Then by Lemma 3.2 N(G, F i ) 2g (2l)g/2 = for any graph G with E(G) = l. Aut(F i ) (2l) V (F i) /2. (5.2) Aut(F i ) (2l) V (F i) /2, (5.3) Now, as F is a spanning subgraph of H, V (F ) = V (H), E(F ) E(H), and so, N(G, H) N(G, F ). Then (5.2) and (5.3) implies q N(G, H) N(G, F ) N(G, F i ) q i= Aut(F i) (2l) V (H) /2 Aut(H) (2l) V (H) /2, i= where the last inequality follows from the fat that any automorphism of H an be restrited to F i to obtain an automorphism of F i, for all i {, 2,..., q}. For the lower bound, let s = 2l and note that, ( ) s N(l, H) N(K s, H) = N(K V (H) V (H), H) thus ompleting the proof. = s V (H) + O(s V (H) ) N(K V (H)! V (H), H) = (2l) V (H) /2 Aut(H) + O(l V (H) /2 /2 ), Remark 5.. Theorem.3 alulates N(l, H) asymptotially exatly, whenever H has a spanning subgraph whih is a disjoint union of yles or isolated edges. The proof also shows that if H is suh a graph then the bound is asymptotially attained by a omplete graph, that is, the omplete graph maximizes the number of H-subgraphs over the set of all graphs with fixed number of edges. However, this is not true for general subgraphs. For example, if the number of edges is l, a omplete graph with 2l verties has O(l 3/2 ) 2-stars, whereas an (l )-star has O(l 2 ) 2-stars. Thus, a omplete graph does not maximize the number of 2-stars for a fixed number of edges. In fat, Alon [3] showed that lim l N(l, H)/l V (H) is finite, and it is non-zero if and only if H is a disjoint union of stars. Moreover, he determined N(l, H) preisely when H is a disjoint union of 2-stars, and also for some other speial types of stars.

15 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS 5 6. Universal Normal Approximation For Uniform Colorings Theorem. says that if E(G n) P, then N(G n ) onverges to infinity as well. Sine a Poisson random variable with mean growing to onverges to a standard normal distribution after standardizing (entering by mean and saling by standard deviation), one possible question of interest is whether N(G n ) properly standardized onverges to a standard normal distribution. Suh a limit theorem an be proved using a diret appliation of Stein method based on exhangeable pairs [3, Theorem ]. However, as before, it turns out that even under the uniform oloring sheme an extra ondition is needed on the struture of the graph for applying it. Nevertheless, as in the Poisson limit theorem of the previous setion, the normality of the standardized random variable N(G n ) is universal and an be proved by a method of moments argument. This setion proves that N(G n ) properly standardized onverges to a standard normal whenever both and E(G n ) / goes to infinity. The alulation of moments in this regime require extensions of Alon s results to multi-graphs and more insights about the exponent γ(h). 6.. Proof of Theorem.2. Let G n G n be a random graph sampled aording to some probability distribution. This setion proves a universal normal limit theorem for ( ) E(Gn ) 2 { Z n := {Y i = Y j } } ( ) E(Gn ) ( 2 = N(G n ) E(G ) n). (i,j) E(G n) Assoiated with every edge of G n define the olletion of random variables {X ij, (i, j) E(G n )}, where X ij are i.i.d. Ber(/), and set ( ) E(Gn ) 2 { W n := X (i,j) } ( ) E(Gn ) ( 2 = M(G n ) E(G ) n). (i,j) E(G n) 6... Comparing Conditional Moments. Begin with two lemmas whih will be used to ompare the onditional moment of Z n and W n. However, unlike in previous setions, non-simple graphs are needed. To this end, define a multi-graph G = (V, E) to be graph where multiple edges are allowed but there are no self loops. For a multi-graph G denote by G S the simple graph obtained from G by removing all multiple edges. A multi-graph H is said to be a multi-subgraph of G if the simple graph H S is a subgraph of G. Observation 6.. Let H = (V (H), E(H)) be a multigraph with no isolated verties. Let F be a multigraph obtained by removing one edge from H and removing all isolated verties formed. Then V (F ) ν(f ) V (H) ν(h). Proof. Observe that ν(f ) ν(h) + and V (H) 2 V (F ) V (H). If V (F ) = V (H) the result is immediate. Now, if V (F ) = V (H), then the vertex removed must have degree and so ν(f ) = ν(h), and the inequality still holds. Finally, if ν(f ) = ν(h) 2, the edge removed must be an isolated edge, in whih ase the number of verties derease by 2 and the number of onneted omponents derease by and the result holds. The above observation helps determine the leading order of the expeted entral moments for multi-subgraph of G n. Lemma 6.. For any multi-subgraph H = (V (H), E(H)) of G n define { Z(H) = {Y i = Y j } }, and X(H) = (i,j) E(H) (i,j) E(H) { X (i,j) }.

16 6 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE Then E(Z(H)) H V (H) ν(h) and E(X(H)) H E(H S ). Proof. By expanding out the produt, Z(H) = E(H) b=0 ( ) b b (i s,j s) E(H), s [ E(H) b] E(H) b s= {Y is = Y js }, (6.) where the seond sum is over all possible hoies of E(H) b distint multi-edges (i, j ), (i 2, j 2 )... (i E(H) b, j E(H) b ) from the multiset E(H). Let F be the subgraph of H formed by (i, j ), (i 2, j 2 )... (i E(H) b, j E(H) b ). Then by Observation 6., V (F ) ν(f ) V (H) ν(h) b, and E(H) b b E {Y is = Y js } = V (F ) ν(f )+b. (6.2) V (H) ν(h) s= As the number of terms in (6.) depends only on H, and for every term (6.2) holds, the result follows. The result for X(H) follows similarly. The leading order of the expetation omes from the first term E X (i,j) = E(H S), (i,j) E(H) and the number of terms depends only on H. The quantity γ(h) was defined for a simple graph by Alon [2]. Friedgut and Kahn [9] showed that γ(h) is the frational stable number of H, whih is the solution of a linear programming problem. Using this alternative definition, we an define γ(h) for any multigraph as follows: γ(h) = arg max φ(v) subjet to φ(x) + φ(y) for (x, y) E(H), φ V H [0,] v V (H) where V H [0, ] is the olletion of all funtions φ : V (H) [0, ]. It is lear that γ(h) = γ(h S ). The polytope defined by the onstraint set of this linear program is alled the frational stable set polytope whih is a well studied objet in ombinatorial optimization [28]. With this definition, we now have the following lemma, whih is a nie result in graph theory in its own right. Lemma 6.2. If for any multi-graph H = (V (H), E(H)) with no isolated verties γ(h) > 2 E(H), then H has a vertex of degree. Moreover, if H is a multi-subgraph of G n whih has a vertex of degree, then E(Z(H) G n ) = E(X(H) G n ) = 0. Proof. Suppose that γ(h) > 2 E(H), and d min(h) 2. Then for any φ : V (H) [0, ] suh that φ(x) + φ(y) for (x, y) E(H), x V (H) whih is a ontradition. φ(x) d min (H) (x,y) E(H) {φ(x) + φ(y)} 2 E(H),

17 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS 7 Now, without loss of generality assume that vertex has degree. Suppose vertex s [n]\{} is the only neighbor of. Therefore, ( E(Z(H) Y, G n ) = E({Y = Y s } Y, G n ) ) { {Y i = Y j } } = 0, (i,j) E(H), (i,j) (,s) whih implies E(Z(H) G n ) = 0. The result for X(H) an be proved similarly. With the above lemmas, the onditional moments of Z n and W n an be ompared. For a simple graph G and a multigraph H define M(G, H) = {G[e, e 2,... e E(H) ] = H}, e E(G) e 2 E(G) e E(H) E(G) where G[e, e 2,... e E(H) ] is the multi-subgraph of G formed by the edges e, e 2,..., e E(H). It is easy to see that M(G, H) H N(G, H S ). Lemma 6.3. Let W n and Z n be as defined before, with and E(G n ) / P, then for every fixed k we have E(Z k n G n ) E(W k n G n ) P 0. Proof. Let M k be the set of all multi-graphs with exatly k multi edges and d min (H) 2. Note that by Lemma 6.2 any H M k must satisfy γ(h) E(H)/2. Expanding the produt and using Lemma 6.2, E(Z k n G n ) = ( E(Gn ) ) k 2 H M k M(G n, H) E(Z(H)), (6.3) By similar argument with Z n replaed by X n, E(W k n G n ) = ( E(Gn ) ) k 2 H M k M(G n, H) E(X(H)), (6.4) Now, let S k M k be the set of all multi-graphs H with d min (H) 2, E(H) = k and γ(h) = E(H) /2 = V (H) ν(h)}. Let ω = E(G n ) /. Now by Lemma 6. and Theorem 5., for any H M k \S k, ( E(Gn ) ) E(H) 2 ( ) E(H) E(Gn ) M(G n, H) E(Z(H)) H H H E(G n ) γ(h) 2 E(H) V (H) ν(h) 2 E(H) ω γ(h) 2 E(H) 2 N(G n, H S ) V (H) ν(h) 0, (6.5) V (H) ν(h) γ(h)

18 8 BHASWAR B. BHATTACHARYA, PERSI DIACONIS, AND SUMIT MUKHERJEE Similarly, for H M k \S k, ( E(Gn ) ) E(H) 2 ( ) E(H) E(Gn ) 2 N(G n, H S ) M(G n, H) E(X(H)) H H ( E(Gn ) ) E(H) The limits in (6.5) and (6.6), together with Equations (6.3) and (6.4) give lim n E(Zk n G n ) E(Wn k G n ) lim n ( E(Gn ) ) k 2 E(H S) 2 N(G n, H S ) 0. (6.6) V (H) ν(h) H S k M(G n, H) E(Z(H)) E(X(H)). Therefore, only multi-subgraphs of G n whih are in S k need to onsidered. As γ(h S ) = γ(h) = V (H) ν(h) = V (H S ) ν(h S ), H S is a disjoint union of stars by Corollary 5.2. Therefore E(H S ) = V (H S ) ν(h S ) = V (H) ν(h) = E(H) /2. This, along with the fat that H annot have any vertex of degree gives that any H S k is a disjoint union of stars, where every edge is repeated twie. Now, it is easy to see that for any suh graph H, E(Z(H)) = E(X(H)), and the result follows from (6.7) Completing the Proof of Theorem.2. To omplete the proof of the normal approximation the following lemma, whih shows that W n satisfies the onditions required in Lemma 8., is needed. D Lemma 6.4. Let W n be as defined before. Then W n N(0, ), and further for any ɛ > 0, t R, ( t k ) lim sup lim sup P k n k! E(W n k G n ) > ɛ = 0. (6.8) Proof. To prove the first onlusion, let T n := M(G n) E(Gn). E(G n) E(Gn) 2 By the Berry-Esseen theorem and the dominated onvergene theorem it follows that T n onverges to N(0, ). Moreover, W n T n P 0, whih implies Wn onverges to N(0, ) by Slutsky s theorem. To prove the seond onlusion, it suffies to show that for any ε > 0, k, (6.7) lim sup P( E(Wn k G n ) > µ k + ε) = 0, (6.9) n where µ k = E(Z k ) and Z is a standard normal random variable. This is beause Ee t Z < for any t, so (6.8) follows by applying Fatou s lemma twie as in the proof of Lemma 3.5. To prove (6.9), note from the proof of Lemma 6.3 that E(W k n G n ) = o P () for odd k. Therefore, assume that k = 2m is even. Reall that S 2m is the number of multi-subgraphs of G n with m

19 UNIVERSAL POISSON AND NORMAL LIMIT THEOREMS IN GRAPH COLORING PROBLEMS 9 double edges, where the underlying simple graph is a disjoint union of stars. Denote by A 2m the olletion of all multi-subgraphs of G n with m double edges. Note that ( ) E(Gn ) (2m)! M(G n, H) M(G n, H) m 2 m, H S 2m H A 2m where in the last step we use the fat that any suh graph in A 2m an be produed by hoosing m out of the E(G n ) edges and then permuting the 2m edges (eah hosen edge doubled) within themselves. Therefore, from the proof of Lemma 6.3 E(W 2m n G n ) = = ( ) E(Gn ) m H S 2m M(G n, H) E(G n ) m ( E(Gn ) m (2m)! 2 m m! + o P (), M(G n, H) E(X(H)) + o P () H S 2m ( ) m + o P () ) (2m)! 2 m E(G n ) m whih establishes (6.9), hene ompleting the proof of Lemma 6.4. ( ) m + o P () Remark 6.. Lemma 6.2 implies that a graph H has a vertex of degree, whenever γ(h) > E(H) /2. In fat, this result is tight, that is, there are graphs, like the yle C g or the omplete bipartite graph K 2,s, with γ(h) = E(H) /2 and no isolated verties. In itself this a nie result in graph theory, whih, to the best of our knowledge, has not been mentioned before. Moreover, it plays a ruial part in our proof Theorem.2. As the number of opies of H in G n is small whenever γ(h) < E(H) /2, these graphs asymptotially do not ontribute to the expetation. The fat that γ(h) > 2 E(H), implies that H has a vertex of degree ensures that the expeted entral moments Z(H) vanish. Therefore, the only graphs that ontribute in the moments are those where γ(h) = E(H) /2. That the threshold E(H) /2, whih is fored from probabilisti alulations, is also the threshold where graphs have degree verties is a surprising oinidene whih illustrates a nie interplay between probability and graph theory in this problem Non-Normal Limit for Fixed Colors. The assumption that the number of olors goes to infinity is essential for the normality in Theorem.2. If E(G n ) / goes to infinity and is fixed, then the limiting distribution of the number of monohromati edges might not be normal, as demonstrated in the following proposition: Proposition 6.. For fixed and the uniform oloring distribution, the number of monohromati edges N(G n ) of the omplete graph K n satisfies: ( ( )) n/ D N(K n ) χ 2 ( ) n 2. Proof. For a [q], define X a be the number of verties of K n with olor a. Then for X = (X, X 2,..., X q ) and p = (/, /,..., /), X Mutlinomial ( n, p ), and n ( ) 2 D X np N(0, Σ),