Subgraph probability of random graphs with specified degrees and applications to chromatic number and connectivity
SSubgraph probability of random graphs with specified degrees andapplications to chromatic number and connectivity
Pu Gao ∗ University of [email protected] Yuval OhapkinUniversity of [email protected]
Abstract
Given a graphical degree sequence d = ( d , . . . , d n ), let G ( n, d ) denote a uniformly randomgraph on vertex set [ n ] where vertex i has degree d i for every 1 ≤ i ≤ n . We give upper andlower bounds on the joint probability of an arbitrary set of edges in G ( n, d ). These upper andlower bounds are approximately what one would get in the configuration model, and thus theanalysis in the configuration model can be translated directly to G ( n, d ), without conditioningon that the configuration model produces a simple graph. Many existing results of G ( n, d ) in theliterature can be significantly improved with simpler proofs, by applying this new probabilistictool. One example we give is about the chromatic number of G ( n, d ).In another application, we use these joint probabilities to study the connectivity of G ( n, d ).When ∆ = o ( M ) where ∆ is the maximum component of d , we fully characterise the connec-tivity phase transition of G ( n, d ). We also give sufficient conditions for G ( n, d ) being connectedwhen ∆ is unrestricted. ∗ Research supported by NSERC. a r X i v : . [ m a t h . C O ] J u l Introduction
Given a graphical degree sequence d = ( d , . . . , d n ), let G ( n, d ) denote a uniformly random graphon vertex set [ n ] where vertex i has degree d i for every 1 ≤ i ≤ n . We also use the same notation G ( n, d ) for the support of G ( n, d ), i.e. the set of graphs with degree sequence d . Unless otherwisespecified, there is usually no confusion from the context whether G ( n, d ) refers to a set or a randomgraph from this set, and we will specify it if there is confusion. We say d is graphical if the set G ( n, d )is nonempty. In the special case where d i = d for every 1 ≤ i ≤ n , G ( n, d ) is a uniformly random d -regular graph. Random graphs with specified degree sequences are among the most studiedrandom graph models. However, unlike the binomial random graph G ( n, p ), edge probabilities in G ( n, d ) are correlated. In fact, even estimating the probability of a single edge in G ( n, d ) can bechallenging. That makes analysing G ( n, d ) difficult. A common tool used for analysing G ( n, d )is the configuration model introduced by Bollob´as [3]. In the configuration model, each vertex isrepresented as a bin containing d i points. Take a uniformly random perfect matching over the total M = (cid:80) ni =1 d i points, and let G ∗ be the multigraph obtained by taking each pair in the matchingas an edge. That is, if points p and q are matched as a pair by the matching then uv is an edgein G ∗ where u contains p and v contains q . G ∗ is a multigraph because there can be more thanone edge between two vertices. A simple counting argument shows that every simple graph withdegree sequence d corresponds to the same number of configurations, and thus, G ∗ , conditionedto the set of simple graphs, has the same distribution as G ( n, d ). Suppose d is such that theprobability that G ∗ is simple is bounded away from 0 for all large n , then any property that holdsa.a.s. in the configuration model must hold a.a.s. in G ( n, d ). Estimating edge probabilities in theconfiguration model is quite easy, and hence, many properties of G ( n, d ) are obtained by analysingthe configuration model. However, the probability that G ∗ is simple is bounded away from 0 onlyfor d such that M = Θ( n ) and M := (cid:80) ni =1 d i = O ( n ). This condition significantly restricts thetype of results one can get by translating from the configuration model.The purpose of this work is to develop probabilistic tools for translating configuration modelanalysis to G ( n, d ) analysis for a rich family of degree sequences. We will illustrate with examplesand show how easy it is to improve existing results and prove new properties of G ( n, d ) using thenew tools we develop. Let H be a graph on [ n ]. Under some mild conditions on H and d , one ofour main results shows the following.The probability that H is a subgraph of G ( n, d ) is approximately the probability that all edges in H appear in the configuration model.Before formally stating the main results, we define a few necessary terms. Given d = ( d , . . . , d n ),without loss of generality we may assume that d ≥ d ≥ · · · ≥ d n . Define∆ = d , δ = d n , M = n (cid:88) i =1 d i , J ( d ) = ∆ (cid:88) i =1 d i . For any S ⊆ [ n ] define d ( S ) = (cid:88) i ∈ S d i , ∆ S = max i ∈ S d i . Given a graph H on vertex set [ n ], let e ( H ) be the number of edges in H , ∆( H ) be the maximumdegree of H , and let d H = ( d H , . . . , d Hn ) denote the degree sequence of H . For graphs on the samevertex set, e.g. on [ n ], we may treat them as sets of edges. Thus, if two graphs H and H are bothon vertex set [ n ], we say H and H are disjoint if their edge sets are disjoint. Given a graph H on[ n ], let H + denote the event that H ⊆ G ( n, d ), i.e. H is a subgraph of G ( n, d ), and let H − denote2he event that H ∩ G ( n, d ) = ∅ . Given two degree sequences d and d (cid:48) defined on the same set ofvertices, we say d (cid:22) d (cid:48) if d i ≤ d (cid:48) i for every i .Typically we consider a sequence of degree sequences indexed by n , and we are interested inproperties of G ( n, d ) asymptotically when n → ∞ . We say a property holds asymptotically almostsurely (a.a.s.) if the probability that the property holds goes to 1 as n → ∞ . We will use standardLandau notation for asymptotic calculations. Given two sequences of real numbers a n and b n ,we say a n = O ( b n ) if there exists a constant C > | a n | ≤ C | b n | for every n . We say a n = o ( b n ) if b n > n ) and lim n →∞ a n /b n = 0. Wewrite a n = Ω( b n ) if a n > b n = O ( a n ). We say a n = ω ( b n ) if a n > b n = o ( a n ). Our first result concerns the conditional edge probabilities in G ( n, d ). Theorem 1.
Let H and H be two disjoint graphs on [ n ] . Suppose that d H (cid:22) d , and uv / ∈ H ∪ H .Then, P ( uv ∈ G ( n, d ) | H +1 , H − ) ≤ ( d u − d H u )( d v − d H v ) M − e ( H ) · f ( d , H , H ) P ( uv ∈ G ( n, d ) | H +1 , H − ) ≥ ( d u − d H u )( d v − d H v ) M − e ( H ) · g ( d , H , H ) where f ( d , H , H ) = (cid:18) − J ( d ) + ∆(8 + d H u + d H v ) M − e ( H ) − e ( H )∆ ( M − e ( H )) + ( d v − d H v )( d u − d H u ) M − e ( H ) (cid:19) − g ( d , H , H ) = (cid:18) − J ( d ) + 6∆ + 2∆( H )∆ M − e ( H ) (cid:19) (cid:18) d v − d H v )( d u − d H u ) M − e ( H ) (cid:19) − . By setting proper conditions on d we obtain an asymptotic value of P ( uv ∈ G ( n, d ) | H +1 , H − )as follows, which is a direct corollary of Theorem 1. Corollary 2.
Let H and H be two disjoint graphs on [ n ] . Suppose that d H (cid:22) d , and uv / ∈ H ∪ H . Suppose further that J ( d ) + ∆ · ∆( H ) = o ( M − e ( H )) , e ( H )∆ = o (cid:0) ( M − e ( H )) (cid:1) . Then, P ( uv ∈ G ( n, d ) | H +1 , H − ) = (1 + o (1)) ( d u − d H u )( d v − d H v ) M − e ( H ) + ( d u − d H u )( d v − d H v ) . Remark 3. If d is a degree sequence composed of i.i.d. copies of a power-law variable with expo-nent between 2 and 3, then a.a.s. J ( d ) = o ( M ) . In [13, Lemma 3], Van der Hofstad, Southwell,Stegehuis and the first author proved the asymptotic conditional probabilities as in Corollary 2 forsuch power-law degree sequences when e ( H ) = O (1) and e ( H ) = 0 . A few other papers stud-ied such conditional probabilities for regular or near-regular degree sequences. Kim, Sudakov andVu [17, Lemma 2.1] obtained the asymptotic value of the conditional probability when e ( H ) = O (1) , e ( H ) = 0 and d i ∼ d for all i where d = o ( n ) and d = ω (1) . D´ıaz, Joos, K¨uhn and Osthus [7, emmas 2.1 and 2.2] obtained the asymptotic value of such conditional probabilities in a random d -regular r -uniform hypergraph. Our corollary above generalises and strengthens all these resultsabove (compared with [7, Lemmas 2.1 and 2.2] for r = 2 , our corollary has more relaxed condi-tions on H and H in addition to relaxed conditions on d ). Our result is mostly comparable withMcKay’s enumeration results [20], which yield similar probabilities as in Corollary 2 but requirestronger conditions on ∆ . These are all for sparse degree sequences. For dense d , such conditionalprobabilities are estimated in [1, 14, 22, 12]. Repeatedly applying Theorem 1 we obtain the following joint probability bounds for an ar-bitrary set of edges, under some mild conditions. Given a graph H , let ∂ ( H ) = { v : v ∩ x (cid:54) = ∅ for some x ∈ E ( H ) } be the set of vertices that are incident to some edge in H . Theorem 4.
Assume H and H are two disjoint graphs on [ n ] .(a) If R ( d , H , H ) := 6 J ( d ) + 2∆(8 + 2∆( H )) M − e ( H ) + 4 e ( H )∆ ( M − e ( H )) ≤ , then P (cid:16) H +1 | H − (cid:17) ≤ n (cid:89) i =1 ( d i ) d H i · h (cid:89) j =1 R ( d , H , H )( M − j + 2) . (b) If r ( d , H , H ) := 2 J ( d ) + 6∆ + 2∆( H )∆ + ∆ ∂ ( H ) M − e ( H ) ≤ , then P (cid:16) H +1 | H − (cid:17) ≥ n (cid:89) i =1 ( d i ) d H i · h (cid:89) j =1 − r ( d , H , H )( M − j + 2) . By setting H = ∅ and proper conditions on d so that R ( d , H , ∅ ) = o (1), we obtain thefollowing useful upper and lower bounds on the joint probability. Corollary 5.
Let H be a graph on [ n ] and assume that d is a degree sequence satisfying J ( d ) = o ( M − e ( H )) . Then P ( H + ) ≤ n (cid:89) i =1 ( d i ) d Hi e ( H ) (cid:89) i =1 o (1) M − i + 2 . (1) If further we have ∆ ∂ ( H ) = o (( M − e ( H ))) then the above holds with equality. Remark 6.
Similar bounds on P ( H + ) are given by McKay in [22, Theorem 2.1] with much morerestrictive d and H . Remark 7.
Although probability estimates similar to (1) appeared in the literature for several times,such probabilities were not connected before to the corresponding probabilities in the configurationmodel, and thus were never used to automatically translate the analysis from the configuration modelto G ( n, d ) . This is one of our main contributions of this work. We compare (1) with the probabilityin the configuration model. Let σ ∗ denote a uniformly random perfect matching over the M pointsproduced by the configuration model, and let G ∗ be the multigraph corresponding to σ ∗ . Given a raph H on [ n ] with d H (cid:22) d , let P ( H ) be the set of matchings σ of size e ( H ) over the set of M pointswhose corresponding graph is H . Then |P ( H ) | = (cid:81) ni =1 ( d i ) d Hi and P ( σ ⊆ σ ∗ ) = (cid:81) e ( H ) i =1 1 M − i +1 .Hence, P ( H ⊆ G ∗ ) ≤ (cid:88) σ ∈P ( H ) P (cid:0) σ ⊆ σ ∗ (cid:1) = n (cid:89) i =1 ( d i ) d Hi e ( H ) (cid:89) i =1 M − i + 1 . (2) Thus, under the condition J ( d ) = o ( M − e ( H )) , our upper bound in (1) differs from the corre-sponding probability bound (2) in the configuration model by a relative o (1) factor in each termof the product. Such an approximation is enough to translate a lot of configuration model analysisto G ( n, d ) . See examples in Sections 1.1.2. Another advantage of using the configuration model is that, it is very easy to bound the proba-bility of having a certain number of edges joining two sets of vertices. Given two subsets of vertices S , S ⊆ [ n ], let e ( S , S ) be the number of edges with one end in S and the other end in S . When S = S , e ( S , S ) is simply the number of edges induced by S . In the configuration model, theprobability that e ( S , S ) ≥ (cid:96) is at most (cid:18) d ( S ) (cid:96) (cid:19) ( d ( S )) (cid:96) (cid:81) (cid:96)i =1 ( M − i + 1) . (3)In the following corollary we show a similar bound for this probability in G ( n, d ). Corollary 8.
Suppose S , S ⊆ [ n ] . Let ≤ (cid:96) < M/ be an integer. Assume that d satisfies J ( d ) = o ( M − (cid:96) ) . Then, P ( e ( S , S ) ≥ (cid:96) ) ≤ (cid:18) d ( S ) (cid:96) (cid:19) ( d ( S )) (cid:96) (cid:32) (cid:96) (cid:89) i =1 o (1) M − i + 2 (cid:33) ≤ (cid:18) d ( S ) (cid:96) (cid:19) ( d ( S )) (cid:96) ( M/ (cid:96) (2 + o (1)) (cid:96) . (4) Remark.
Since J ( d ) ≤ ∆ , both Corollaries 5 and 8 hold when J ( d ) is replaced by ∆ . Proof.
Let H (cid:96) denote the set of graphs on [ n ] having exactly (cid:96) edges, all of which have exactlyone end in S and the other end in S . Let { i , · · · , i k } denote the set of vertices in S ∪ S . Bythe union bound and Corollary 5, P ( e ( S , S ) ≥ (cid:96) ) ≤ (cid:88) H ∈H (cid:96) P ( H ⊆ G ( n, d )) ≤ (cid:32) (cid:96) (cid:89) i =1 o (1) M − i + 2 (cid:33) (cid:88) H ∈H (cid:96) k (cid:89) j =1 ( d i j ) d Hij . Next we give a combinatorial interpretation of (cid:80) H ∈H (cid:96) (cid:81) kj =1 ( d i j ) d Hij above. Represent each vertex i j in S ∪ S by a bin containing d i j points. Given H ∈ H (cid:96) , how many matchings of size (cid:96) over 2 (cid:96) out of the total d ( S ∪ S ) points are there so that if uv is an edge in H then there is a point p in bin u and a point q in bin v such that p and q are matched by the matching? It is easy to seethat there are exactly (cid:81) kj =1 ( d i j ) d Hij such matchings. Hence, (cid:80) H ∈H (cid:96) (cid:81) kj =1 ( d i j ) d Hij is bounded aboveby the total number of size- (cid:96) matchings where every pair ( p, q ) in the matching is of the form that p is in some bin in S and q is in some bin in S . There are (cid:0) d ( S ) (cid:96) (cid:1) ways to choose the (cid:96) ends thatare in bins in S and there are ( d ( S )) (cid:96) ways to choose the other ends from bins in S and matchthem to the (cid:96) ends chosen before. Hence, (cid:88) H ∈H (cid:96) k (cid:89) j =1 ( d i j ) d Hij ≤ (cid:18) d ( S ) (cid:96) (cid:19) ( d ( S )) (cid:96) , P ( e ( S , S ) ≥ (cid:96) ) ≤ (cid:32) (cid:96) (cid:89) i =1 o (1) M − i + 2 (cid:33) (cid:18) d ( S ) (cid:96) (cid:19) ( d ( S )) (cid:96) = (cid:18) d ( S ) (cid:96) (cid:19) ( d ( S )) (cid:96) ( M/ (cid:96) (2 + o (1)) (cid:96) , as (cid:81) (cid:96)i =1 ( M − i + 2) = ( M/ (cid:96) (cid:96) . G ( n, d )Let χ ( G ) denote the chromatic number of graph G , i.e. the minimum number of colours required tocolour vertices of G so that all pairs of adjacent vertices receive distinct colours. It is known thata.a.s. the chromatic number of a random d -regular graph is asymptotically d/ d , for d = ω (1)and d = o ( n ); see [11, 5, 18]. In the paper [10] Frieze, Krivelevich and Smyth asked under whatconditions on d would we have a.a.s. χ ( G ( n, d )) = Θ( d/ ln d ), where d = M/n is the average degreeof graphs in G ( n, d ). Let D k = (cid:80) ki =1 d i and M = (cid:80) ni =1 d i . It was shown in [10] that if(A1) there exist constants 1 / < α < (cid:15), K > D k ≤ K dn ( k/n ) α for all 1 ≤ k ≤ (cid:15)n ;(A2) ∆ = o ( M ),then a.a.s. χ ( G ( n, d )) = O ( d/ ln d ). On the other hand, if(A3) ∆ = o ( M ),then a.a.s. χ ( G ( n, d )) = Ω( d/ ln d ).We significantly relax the conditions (A2) and (A3) and obtain the following result for χ ( G ( n, d )). Theorem 9. If d satisfies condition (A1) and ∆ = o ( n ) then a.a.s. χ ( G ( n, d )) = O ( d/ ln d ) . If J ( d ) = o ( M ) then a.a.s. χ ( G ( n, d )) = Ω( d/ ln d ) . The proof of Theorem 9 will be given in Section 2, which is obtained by translating the existinganalysis [10] from the configuration model to G ( n, d ). We believe that many other results of G ( n, d )can be obtained or improved in a similar manner. For instance, the order of the largest component of G ( n, d ) was determined by Molloy and Reed [23] for the so-called “well-behaved” degree sequences.Their proof relies on an analysis in the configuration model. We believe that most of the analysiscan be immediately translated to G ( n, d ) by using the conditional edge probabilities in Theorem 1.We also believe that the new probabilistic tools developed in Section 1.1.1 will be useful in studyingother properties of G ( n, d ). We give another example in Section 1.1.3. G ( n, d )The connectivity is one of the best studied graph properties for random graphs. Erd˝os and R´enyi [6]determined the threshold of the connectedness for G ( n, p ). Indeed, for every fixed integer k ≥ G ( n, p ) becomes k -connected. Random graph G ( n, p ) becomesconnected when isolated vertices disappear, which happens when p ≈ log n/n . For p in this range,the average degree of the random graphs is around log n and there are very few vertices of degreeone or two, and these vertices are pair-wise far away in graph distance. Consequently the verticesof degree one or two do not affect the connectedness of G ( n, p ). The most natural sparser randomgraph model for the study of connectivity would be G ( n, d ), and it is natural to ask when are suchrandom graphs connected. As we will see, the vertices of degree one or two play crucial roles forthe connectedness of G ( n, d ). 6he first work about the connectivity of G ( n, d ) was by Wormald [24] in 1981. In this pioneeringwork, the author studied the connectivity of G ( n, d ) where δ ≥ R (i.e. R does not depend on n ), and proved that a.a.s. the connectivity of G ( n, d ) is equal to δ for such degree sequences. Frieze [9] studied the connectivity of random d -regular graphs where 3 ≤ d = o ( n . ), and proved that a.a.s. a random d -regular graph is d -connected, for d in the aforementioned range. Later, Cooper, Frieze and Reed [4] extended thisresult to 3 ≤ d ≤ cn where c > δ = 2. (cid:32)Luczakshowed that, for any d where δ ≥ ≤ n . then a.a.s. G ( n, d ) is δ -connected. When δ = 2and ∆ ≤ n . he characterised the structure of G ( n, d ) and determined when is G ( n, d ) a.a.s.2-connected. In a more recent work, Federico and Van der Hofstad [8] considered degree sequencespermitting δ = 1 and fully charactersized the connectivity transition of G ( n, d ) for d ∈ D , where D = { d : M = Θ( n ) , (cid:80) ni =1 d i = O ( n ) } . Let n be the number of components in d with value1, and n the number of components in d with value 2. Federico and Van der Hofstad showedthat for d ∈ D that satisfies some additional “smoothness” condition, G ( n, d ) is a.a.s. connectedif n = o ( √ n ) and n = o ( n ), and G ( n, d ) is disconnected if n = ω ( √ n ). All the work that wehave discussed so far are for d where either the maximum degree is not large (at most n . ), or d corresponds to a regular degree sequence, and the degree is nearly sublinear (at most cn for somesufficiently small c ). For d linear in n , Krivelevich, Sudakv, Vu and Wormald [18] proved severalproperties of random d -regular graphs, including the connectivity. Recently, Isaev, McKay and thefirst author [12] proved several properties including the connectivity of G ( n, d ) for near-regular d where d = ω (log n ) and d i ∼ d for every i .In this work, we characterise the connectivity transition of G ( n, d ) for a much larger family ofdegree sequences. For the family of d where J ( d ) = o ( M ) (in particular, when ∆ = o ( M )) wefully characterise the phase transition of the connectedness of G ( n, d ). When ∆ is unrestricted wegive sufficient conditions under which G ( n, d ) is a.a.s. connected.We only consider degree sequences where δ ≥ G ( n, d ) is disconnected trivially.Given the degree sequence d where ∆ = d ≥ d ≥ · · · ≥ d n = δ ≥
1, define n = n (cid:88) i =1 { d i =1 } , n = n (cid:88) i =1 { d i =2 } . Theorem 10.
Assume d is a graphical degree sequence with J ( d ) = o ( M ) . Let c > be a fixedconstant.(a) If n = o ( √ M ) and n = o ( M ) then a.a.s. G ( n, d ) is connected.(b) If n = ω ( √ M ) then a.a.s. G ( n, d ) is disconnected.(c) If n ≥ c √ M or n ≥ c √ M then there exists δ = δ ( c ) > such that for all sufficiently large n , P ( G ( n, d ) disconnected ) ≥ δ. Since J ( d ) ≤ ∆ , we immediately have the following corollary. Corollary 11.
Theorem 10 holds if J ( d ) = o ( M ) is replaced by ∆ = o ( M ) . Next, we deal with degree sequences where ∆ is rather large. Define H = { i : d i ≥ √ M / log M } . Our next result gives sufficient conditions for the connectedness of G ( n, d ).7 heorem 12. Assume M − d ( H ) = Ω( M ) . If n = o ( √ M ) and n = o ( M ) then a.a.s. G ( n, d ) isconnected. Conditions in Theorem 12 are not necessary. We can easily make up d where d = n − M = Θ( n ) and a linear number of vertices have degree 1 (or of degree 2). Conditions in Theorem 12are not satisfied but G ( n, d ) is always connected for such degree sequences. J ( d ) = o ( M ) A rich family of degree sequences satisfies the condition J ( d ) = o ( M ) for which we may apply theprobabilities in Theorems 1 and 4. For instance, it covers all regular sublinear degree sequences,i.e. d i = d for all i and d = o ( n ). We give two additional examples below that might be of interestin applications. • ∆ = o ( n ) and a linear (in n ) number of vertices have degree Ω(∆). • d is composed of i.i.d. power-law variables of exponent τ >
2, conditioned to even sum.The reader may wonder what types of degree sequences do not satisfy J ( d ) = o ( M ). Certainly,regular degree sequences with linear degrees do not satisfy this condition. More generally, if thereis a linear (in n ) number of vertices with degree Θ( n ), then that degree sequence does not satisfy J ( d ) = o ( M ).We will prove Theorems 1 and 4 in Section 3. The proof of Theorem 9 will be given in Section 2and the proofs for Theorems 10 and 12 will be presented in Section 4. We first briefly sketch the proof in [10]. For the upper bound, the authors first obtained an O ( d/ ln d ) upper bound for the multigraph G ∗ from the configuration model. A multigraph isproperly coloured if every pair of adjacent and distinct vertices are coloured differently. OnlyCondition (A1) is needed for this part of the proof. Then they applied a sequence of switchingoperations which repeatedly switch away the loops and multiple edges in G ∗ . Then they provedthat (a), every simple graph is obtained with asymptotically the same probability after applyingthe switchings; (b) if H is the graph induced the by set of edges added during the switchings, thena.a.s. χ ( H ) = O (1). It follows immediately that a.a.s. the chromatic number of G ( n, d ) is at most χ ( G ∗ ) · χ ( H ) = O ( d/ ln d ). Condition (A2) is needed to guarantee (a) and (b).For the lower bound, they proved that for any partition σ of vertices in to t = b · d/ ln d parts,where b > σ specifies a proper t -colouring of G ∗ is at most t − n . Condition (A3) was applied to obtain a lower bound on the probability that G ∗ is a simple graph. When ∆ = o ( n ), the probability t − n is small enough to beat the unionbound over all such partitions σ , and the inverse of the probability that G ∗ is simple. Note that(A3) implies ∆ = o ( n ).To prove Theorem 9, we carry all analysis from [10] for G ∗ to G ( n, d ). Both the upper andlower bound proofs for χ ( G ∗ ) in [10] use upper bounds on the probability of G ∗ containing someset of edges H where M − e ( H ) = Ω( M ). The upper bound of χ ( G ∗ ) follows by [10, Lemmas1–3]. These lemmas hold for G ( n, d ) with exactly the same proofs, by applying inequalities (1)and (4) instead of (2) and (3). The additional 1 + o (1) factors in (1) and (4), compared with (2)and (3), do not affect the proof (in fact, any constant factor would do). As no switching analysisis required any more, we do not need condition (A2). Instead, we need J ( d ) = o ( M ) in order to8pply (1) and (4). This is guaranteed by our assumption (A1) and ∆ = o ( n ) as follows: by (A1), J ( d ) /M = O ((∆ /n ) α ) which is o (1) if ∆ = o ( n ).The same translation of analysis holds for the lower bound proof. As we are working on G ( n, d )instead of G ∗ , it is sufficient if the probability t − n beats the union bound over the total number ofsuch partitions. This is always the case as there can be at most t n partitions into t parts. Hence,for the lower bound we do not need conditions (A3) or ∆ = o ( n ) any more. Instead, we impose J ( d ) = o ( M ) which validates the application of the probability bounds (1) and (4). Proof of Theorem 4.
Let e = u v , . . . , e h = u h v h be an enumeration of the set of edges in H where h = e ( H ). Let G = ∅ and G j = G j − ∪ { e j } for every 1 ≤ j ≤ h . Then, P (cid:16) H +1 | H − (cid:17) = h (cid:89) j =1 P ( e j | G + j − , H − ) . By Theorem 1, h (cid:89) j =1 P ( e j | G + j − , H − ) ≤ h (cid:89) j =1 ( d u j − d G j − u j )( d v j − d G j − v j ) f ( d , G j − , H ) M − j + 2= n (cid:89) i =1 ( d i ) d H i · h (cid:89) j =1 f ( d , G j − , H )( M − j + 2) , where f ( d , G j − , H ) ≤ (cid:18) − J ( d ) + ∆(8 + 2∆( H )) M − j + 2 − e ( H )∆ ( M − j + 2) (cid:19) − ≤ J ( d ) + 2∆(8 + 2∆( H )) M − j + 2 + 4 e ( H )∆ ( M − j + 2) = 1 + R ( d , H , H ) , for every 1 ≤ j ≤ h . The second inequality above holds by the fact that (1 − x ) − ≤ x for all x ∈ [0 , /
2] and the assumption that R ( d , H , H ) ≤
1. This yields our upper bound on P ( H +1 | H − ). Again by the lower bound in Theorem 1, h (cid:89) j =1 P ( e j | G + j − , H − ) ≥ h (cid:89) j =1 ( d u j − d G j − u j )( d v j − d G j − v j ) g ( d , G j − , H ) M − j + 2= n (cid:89) i =1 ( d i ) d H i · h (cid:89) j =1 g ( d , G j − , H )( M − j + 2) , where g ( d , G j − , H ) = (cid:18) − J ( d ) + 6∆ + 2∆( H )∆ M − j + 2 (cid:19) (cid:32) d v j − d G j − v j )( d u j − d G j − u j ) M − j + 2 (cid:33) − ≥ − J ( d ) + 6∆ + 2∆( H )∆ + d v j d u j M − j + 2 ≥ − r ( d , H , H ) , vx ya b u vx ya bG G Figure 1: Forward switchingfor every 1 ≤ j ≤ h . The second inequality above holds by the fact that (1 + x ) − ≥ − x for all x ≥
0. This yields our lower bound on P ( H +1 | H − ). Proof of Theorem 1.
Let G denote the set of graphs G on [ n ] with degree sequence d , . . . , d n ,such that H ⊆ G , G ∩ H = ∅ . Let S denote the set of graphs in G that contain uv as an edge,and let S = G \ S . Then, P (cid:0) uv ∈ G ( n, d ) | H +1 , H − (cid:1) = | S || S | + | S | = 11 + | S | / | S | We will obtain upper and lower bounds on the ratio | S | / | S | by analysing switchings that relategraphs in S to graphs in S . We first define the switching.Given G ∈ S , a forward switching specifies an ordered 4-tuple ( x, a, y, b ) ∈ [ n ] satisfying thefollowing conditions:(1) u, v, x, y, a, b are all distinct, except x = y is permitted.(2) xa and yb are edges in G \ H .(3) None of xu , yv , and ab are edges in G ∪ H .Then the forward switching converts G to a graph G (cid:48) ∈ S by deleting the edges uv , xa and yb from G and adding the edges xu , yv and ab . See Figure 1 for an illustration, where solid lines denoteedges in the graph and dashed lines denote non-edges.Let f ( G ) denote the number of forward switchings that can be applied to G . We will show thefollowing upper and lower bounds on f ( G ): Claim 13. ( a ) f ( G ) ≤ ( M − e ( H )) ( b ) f ( G ) ≥ ( M − e ( H )) (cid:18) − J ( d ) + ∆(8 + d H u + d H v ) M − e ( H ) − e ( H )∆ ( M − e ( H )) (cid:19) . Next, given G (cid:48) ∈ S , we count the number of forward switchings that can produce G (cid:48) . In orderto do so, we define a backward switching on G (cid:48) as an ordered 4-tuple ( x, a, y, b ) ∈ [ n ] satisfyingthe following:(1’) u, v, x, y, a, b are all distinct, except x = y is permitted.102’) xa and yb are not edges in G (cid:48) ∪ H .(3’) xu , yv , and ab are edges in G (cid:48) \ H .Then the backward switching deletes the edges xu , yv , and ab , and adds the edges uv , xa and yb .Obviously, a backward switching on G (cid:48) is exactly the inverse of a forward switching whichproduces G (cid:48) . Let b ( G (cid:48) ) be the number of backward switchings that can be applied to G (cid:48) . We willshow the following. Claim 14. ( a ) b ( G (cid:48) ) ≤ ( d u − d H u )( d v − d H v )( M − e ( H ))( b ) b ( G (cid:48) ) ≥ ( d u − d H u )( d v − d H v )( M − e ( H )) (cid:18) − J ( d ) + 6∆ + 2∆( H )∆ M − e ( H ) (cid:19) . Let T be the total number of forward switchings from S to S . By definition, T = (cid:88) G ∈ S f ( G ) = (cid:88) G (cid:48) ∈ S b ( G (cid:48) ) . By Claim 13(a) and Claim 14(b), | S | ( d u − d H u )( d v − d H v )( M − e ( H )) (cid:18) − J ( d ) + 6∆ + 2∆( H )∆ M − e ( H ) (cid:19) ≤ T ≤ | S | ( M − e ( H )) . Thus, | S || S | + | S | ≥ ( d v − d H v )( d u − d H u ) M − e ( H ) (cid:18) − J ( d ) + 6∆ + 2∆( H )∆ M − e ( H ) (cid:19) (cid:18) d v − d H v )( d u − d H u ) M − e ( H ) (cid:19) − = ( d v − d H v )( d u − d H u ) M − e ( H ) · g ( d , H , H ) . Similarly, Claim 13(b) and Claim 14(a), | S || S | + | S | ≤ ( d v − d H v )( d u − d H u ) M − e ( H ) · f ( d , H , H ) . Hence, we have shown the upper and lower bounds of P ( uv ∈ G ( n, d ) | H +1 , H − ) as desired.It only remains to prove the two claims. They follow from simple inclusion-exclusion countingarguments as follows. Proof of Claim 13.
The upper bound is obvious as there are at most M − e ( H ) ways tochoose vertices x and a , and then at most M − e ( H ) ways to choose vertices y and b . To get therequired lower bound, we subtract from the above upper bound the number of choices where oneof the conditions in (1)–(3) is violated. If condition (1) is violated, then { x, y, a, b } ∩ { u, v } (cid:54) = ∅ , or a ∈ { y, b } , or x = b . There are at most 2( M − e ( H )) · · M − e ( H )) + ∆( M − e ( H )) =11∆( M − e ( H )) ways to choose such 4-tuples. In our upper bound, we only considered choiceswhere condition (2) is satisfied. Thus, it only remains to subtract the number of choices wherecondition (3) is violated. That means either (a), xu , yv , or ab is an edge in G ; or (b), xu , yv , or ab is an edge in H . We call an ordered triple of vertices ( v , v , v ) a directed 2-path at v , if both v v and v v are edges in the graph. Note that for any graph G with degree sequence d , and any11 ∈ [ n ], the number of directed 2-paths at v in G is always at most (cid:80) ∆ i =1 ( d i −
1) = J ( d ) − ∆.Hence, the number of choices for (a) is at most 3( J ( d ) − ∆)( M − e ( H + )). The number of choicesfor (b) is at most ( d H u ∆ + d H v ∆)( M − e ( H )) + 2 e ( H )∆ . These give the lower bound for f ( G )as desired. Proof of Claim 14.
The upper bound is obvious. There are at most d u − d H u ways to choose x , at most d v − d H v ways to choose y , and at most M − e ( H ) ways to choose a and b . From thisupper bound, we need to subtract the number of choices where condition (1’) or (2’) is violated(note that our choices in the upper bound guarantee condition (3’) already). If condition (1’) isviolated then { a, b } ∩ { u, v, x, y } (cid:54) = ∅ . There are at most 4 · · ( d u − d H u )( d v − d H v )∆ such choices.If condition (2’) is violated then either (a’), xa or yb is an edge in G (cid:48) ; or (b’), xa or yb is an edgein H . The number of choices for (a’) is at most 2( d u − d H u )( d v − d H v )( J ( d ) − ∆), and the numberof choice for (b’) is at most 2( d u − d H u )( d v − d H v )∆( H )∆. Subtracting these upper bounds onthe number of invalid choices from the upper bound on b ( G (cid:48) ) yields the lower bound on b ( G (cid:48) ) asdesired. Approximately four proof techniques or a hybrid of them have been used for proving the connectivityof G ( n, d ) and for analysing properties of G ( n, d ) in general, when ∆ is not too large. The first,and perhaps the most well known method uses the configuration model. Recall that all a.a.s.results can be translated from the configuration model to G ( n, d ) if d ∈ D where D = { d : M =Θ( n ) , (cid:80) ni =1 d i = O ( n ) } . Federico and Hofstad’s work [8] is an example of this proof method. Dueto the ease in handling with the configuration model, they managed to prove more accurate resultincluding a critical window analysis during the connectivity phase transition, but such distributionalresults cannot be directly translated to G ( n, d ).Another proof method uses graph enumeration. Assume we want to bound the probability that e ( S, S ) = 0. This probability is simply |G ( S, d | S ) | · |G ( S, d | S ) ||G ( n, d ) | , where d | S denotes the degree sequence obtained by restricted to vertices in S , and G ( S, d | S ) denotethe set of graphs on vertex set S and with degree sequence d | S . Applying known asymptoticenumeration results on |G ( n, d ) | one can get asymptotic probability for the event that e ( S, S ),which can further be used to bound the probability that G ( n, d ) is disconnected. This approachwas taken by Wormald [24]. It may be interesting to note that the enumeration results on whichWormald’s proof was based are by Bender and Canfield [2], which requires ∆ to be absolutelybounded. Then Bollob´as introduced the configuration model and deduced a probabilistic proofof [2]. Afterwards the configuration model became popularised. In that sense, Wormald’s proofcan be viewed as a “detour” of the first method aforementioned.The third method combines the configuration model with the switching method introduced byMcKay [20, 21]. As mentioned before, a.a.s. results can be translated from the configuration modelto G ( n, d ) if d ∈ D . What can we do for d / ∈ D , e.g. when the average degree of d is growingwith n ? McKay’s switching method starts with G ∗ , the multigraph produced by the configurationmodel, and repeatedly switches away multiple edges in the multigraph from the configuration model.Using simple counting argument one can show that when ∆ is below M / then the distribution12f the final simple graph obtained is very close to the uniform distribution. Then, we can deduceproperties of G ( n, d ) by analysing the configuration model and the switching algorithm. There aremany results of G ( n, d ) obtained this way, e.g. χ ( G ( n, d )) in [10] discussed in Section 2. See moreexamples in [15]. In terms of the connectivity, proofs in [9, 19] followed this path.The last method applies the switching technique directly to random graph G ( n, d ). Partitionthe set of graphs G ( n, d ) into two parts S and T where graphs in S have a certain property P andgraphs in T do not. Then defining switchings that relate graphs in S to graphs in T . By countingthe number of ways to perform switchings one can estimate the ratio | S | / | T | and the probabilityof property P . This approach was used in [4] for the connectivity of random d -regular graphs for d up to cn where c is sufficiently small.In this work, we use the new tool in Corollary 5 to characterise the connectivity phase transitionfor the family of degree sequences where J ( d ) = o ( M ) (Theorem 10). This result is a generalisationof [8] but works for a much larger family of degree sequences. For Theorem 12 we will use switchingsto prove that the set of edges incident with H spans a subgraph with O (1) components. Then, weexpose the set of edges incident with H , and then analyse the subgraph induced by [ n ] \ H . As thedegrees of vertices in [ n ] \ H are not too large, we can apply Corollary 5 again. Let Y denote the number of isolated edges, i.e. edges whose ends are both of degree 1. Let Z be thenumber of isolated triangles, i.e. triples of vertices { x, y, z } who induce a K and all of the threevertices are of degree 2. With standard first and second moment calculations using the asymptoticprobabilities in Corollary 5 we immediately have the following lemma, whose proof we omit. Lemma 15. • If n = Ω( √ M ) then E Y ∼ n / M and E Y ( Y − ∼ n / M . • If n = Ω( M ) then E Z ∼ n / M and E Z ( Z − ∼ n / M . Now Theorem 10(b) follows by Chebyshev’s inequality, and Theorem 10(c) follows by the Paley-Zygmund inequality.
Proposition 16.
For any d with even sum, |G ( n, d ) | ≤ M !2 M/ ( M/ (cid:81) i ∈ [ n ] d i ! . Proof.
Represent vertex i by a bin containing exactly d i points. A perfect matching over the total M points in the n bins is called a pairing . A pairing produces a multigraph with degree sequence d by representing each { v ( p ) , v ( q ) } as an edge where p and q are points matched by the pairingand v ( p ) and v ( q ) are the bins/vertices that contain points p and q respectively. It is easy to seethat every graph in G ( n, d ) corresponds to exactly (cid:81) i ∈ [ n ] d i ! pairings. On the other hand, thereare exactly M ! / M/ ( M/ M points. The assertion follows.Given S ⊆ [ n ] let X S denote the indicator variable for the event that e ( S, S ) = 0. Recall that d ( S ) = (cid:80) i ∈ S d i and d | S = ( d i ) i ∈ S . Lemma 17.
Assume J ( d ) = o ( M ) . Suppose S ⊆ [ n ] where M − d ( S ) = Ω( M ) . Then, E X S ≤ ( √ o (1)) (cid:18) (1 + o (1)) d ( S ) M − d ( S ) (cid:19) d ( S ) / (cid:18) − d ( S ) M (cid:19) M/ . roof. By Proposition 16, the number of graphs on S with degree sequence d | S is at most d ( S )!2 d ( S ) / ( d ( S ) / (cid:81) i ∈ S d i ! . By (1), E X S ≤ d ( S )!2 d ( S ) / ( d ( S ) / (cid:81) i ∈ S d i ! · (cid:89) i ∈ S d i ! d ( S ) / (cid:89) i =1 o (1) M − i ≤ ( √ o (1)) d ( S )!2 d ( S ) / ( d ( S ) / · (2 + o (1)) − d ( S ) / (( M − d ( S )) / M/ ≤ ( √ o (1)) (cid:18) (1 + o (1)) d ( S ) M − d ( S ) (cid:19) d ( S ) / (cid:18) − d ( S ) M (cid:19) M/ . If G ( n, d ) is disconnected then there is a component of G ( n, d ) with total degree at most M/ S ⊆ [ n ] where d ( S ) ≤ M/ e ( S, S ) = 0. Inthe next lemma, we first bound the expected number of such S where d ( S ) ≥ . | S | . Lemma 18.
A.a.s. there are no nonempty sets S ⊆ [ n ] where . | S | ≤ d ( S ) ≤ M/ and e ( S, S ) = 0 . Proof.
Let (cid:15) = 0 .
5. Suppose S is a set of vertices with d ( S ) = h ≤ M/ d ( S ) ≥ (2 + (cid:15) ) | S | .Then, | S | ≤ h/ (2 + (cid:15) ). Hence, by Lemma 17, for all sufficiently large n , (cid:88) S ⊆ [ n ]: d ( S )= hh ≥ (2+ (cid:15) ) | S | E X S ≤ (cid:18)(cid:18) nh/ (2 + (cid:15) ) (cid:19) { h (cid:15) < n } + 2 n { h (cid:15) ≥ n } (cid:19) (cid:18) (1 + o (1)) ρ − ρ (cid:19) h/ (1 − ρ ) M/ , where ρ = h/M < /
2. By the assumption that n = o ( √ M ) and n = o ( M ), we must have M ≥ (3 − o (1)) n . Thus the above is at most2 (cid:32) (2 e (1 + (cid:15) ) / / (2+ (cid:15) ) ρ (cid:15)/ (2+ (cid:15) ) − ρ (cid:33) h/ (1 − ρ ) M/ + 2(2 − (cid:15)/ ρ/ (1 − ρ )) h/ (1 − ρ ) M/ . We prove that (cid:88) ≤ h ≤ M/ (cid:32) (2 e (1 + (cid:15) ) / / (2+ (cid:15) ) ρ (cid:15)/ (2+ (cid:15) ) − ρ (cid:33) h/ (1 − ρ ) M/ = o (1) , and (cid:88) ≤ h ≤ M/ (2 − (cid:15)/ ρ/ (1 − ρ )) h/ (1 − ρ ) M/ = o (1) , which will complete the proof of the lemma. Note that (cid:88) ≤ h ≤ M/ (cid:32) (2 e (1 + (cid:15) ) / / (2+ (cid:15) ) ρ (cid:15)/ (2+ (cid:15) ) − ρ (cid:33) h/ (1 − ρ ) M/ = (cid:88) ≤ h< ln n (cid:16) O (1)(ln n/M ) (cid:15)/ (2+ (cid:15) ) (cid:17) h/ + (cid:88) ln n ≤ h< . M . h/ + (cid:88) . M ≤ h ≤ M/ exp (cid:0) f ( ρ ) M/ (cid:1) , f ( ρ ) = 2 ρ (cid:15) ln(2 e (1 + (cid:15) ) /
3) + (cid:15)ρ (cid:15) ln( ρ ) + (1 − ρ ) ln(1 − ρ ) . The function f ( ρ ) is below − .
01 uniformly over ρ ∈ [0 . , . (cid:88) ≤ h ≤ M/ (cid:32) (2 e (1 + (cid:15) ) / / (2+ (cid:15) ) ρ (cid:15)/ (2+ (cid:15) ) − ρ (cid:33) h/ (1 − ρ ) M/ = o (1) . Bounding (cid:80) ≤ h ≤ M/ (2 − (cid:15)/ ρ/ (1 − ρ )) h/ (1 − ρ ) M/ by o (1) can be done in a similar manner.We are ready to complete the proof for part (a) of Theorem 10. Proof of Theorem 10(a).
By Lemma 18 it only remains to show that a.a.s. there are nononempty sets S ⊆ [ n ] where d ( S ) ≤ min { . | S | , M/ } and e ( S, S ) = 0. Let ξ = ξ n = o (1) be suchthat n / √ M ≤ ξ and n /M ≤ ξ . Let S be a subset of vertices with (cid:96) vertices of degree 1, (cid:96) verticesof degree 2, and (cid:96) ≥ vertices of degree at least 3, d ( S ) ≤ M/ d ( S ) ≤ . | S | . The number ofways to choose such a set S is at most (cid:0) n (cid:96) (cid:1)(cid:0) n (cid:96) (cid:1)(cid:0) n(cid:96) ≥ (cid:1) . Given such an S , d ( S ) ≥ (cid:96) + 2 (cid:96) + 3 (cid:96) ≥ . Itfollows immediately that (cid:96) + 2 (cid:96) + 3 (cid:96) ≥ ≤ . (cid:96) + (cid:96) + (cid:96) ≥ ) , which implies that (cid:96) ≥ ≤ (cid:96) + (cid:96) , (cid:96) + 2 (cid:96) + 3 (cid:96) ≥ ≤ d ( S ) ≤ . (cid:96) + (cid:96) + (cid:96) ≥ ) ≤ (cid:96) + 5 (cid:96) . (5)In the rest of the proof, for simplicity we use C to denote an absolute positive constant, which maytake different values at different places where, the actual values of the constants do not matter. ByLemma 17, the probability that e ( S, S ) = 0 is at most (cid:18) Cd ( S ) M (cid:19) d ( S ) / ≤ (cid:18) C ( (cid:96) + (cid:96) ) M (cid:19) (cid:96) / (cid:96) +3 (cid:96) ≥ / . Hence, the expected number of sets S where d ( S ) ≤ . | S | , d ( S ) ≤ M/ e ( S, S ) = 0 is at most (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:18) n (cid:96) (cid:19)(cid:18) n (cid:96) (cid:19)(cid:18) n(cid:96) ≥ (cid:19) (cid:18) C ( (cid:96) + (cid:96) ) M (cid:19) (cid:96) / (cid:96) +3 (cid:96) ≥ / ≤ (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:32) Cn (cid:96) (cid:114) (cid:96) + (cid:96) M (cid:33) (cid:96) (cid:18) Cn ( (cid:96) + (cid:96) ) (cid:96) M (cid:19) (cid:96) (cid:32) Cn ( (cid:96) + (cid:96) ) / (cid:96) ≥ M / (cid:33) (cid:96) ≥ ≤ (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:18) Cξ √ (cid:96) + (cid:96) (cid:96) (cid:19) (cid:96) (cid:18) Cξ (cid:96) + (cid:96) (cid:96) (cid:19) (cid:96) (cid:32) C ( (cid:96) + (cid:96) ) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ . We split the above sum into two parts, one restricted to (cid:96) ≥ (cid:96) and the other restricted to (cid:96) < (cid:96) ,and we show that each sum is o (1). Suppose (cid:96) ≥ (cid:96) . Then (cid:96) + (cid:96) ≤ (cid:96) . Hence, (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:96) ≥ (cid:96) (cid:18) Cξ √ (cid:96) + (cid:96) (cid:96) (cid:19) (cid:96) (cid:18) Cξ (cid:96) + (cid:96) (cid:96) (cid:19) (cid:96) (cid:32) C ( (cid:96) + (cid:96) ) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ ≤ (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:96) ≥ (cid:96) (cid:18) Cξ √ (cid:96) (cid:19) (cid:96) (cid:18) Cξ (cid:96) (cid:96) (cid:19) (cid:96) (cid:32) C(cid:96) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ . g ( x ) = ( K/x ) x on x ≥ K >
0. By considering the derivative of ln( g ( x )), it is easy tosee that g ( x ) is maximised at x = K/e . Thus, (cid:18)
Cξ (cid:96) (cid:96) (cid:19) (cid:96) ≤ exp (cid:18) Cξ(cid:96) e (cid:19) , and (cid:32) C(cid:96) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ ≤ exp (cid:32) C(cid:96) / e √ M (cid:33) . It follows now that (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:96) ≥ (cid:96) (cid:18) Cξ √ (cid:96) (cid:19) (cid:96) (cid:18) Cξ (cid:96) (cid:96) (cid:19) (cid:96) (cid:32) C(cid:96) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ = (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:96) ≥ (cid:96) (cid:96) ≤ M / / log M (cid:18) Cξ √ (cid:96) (cid:19) (cid:96) (cid:18) Cξ (cid:96) (cid:96) (cid:19) (cid:96) (cid:32) C(cid:96) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ + (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:96) ≥ (cid:96) M / / log M<(cid:96) ≤ n (cid:18) Cξ √ (cid:96) (cid:19) (cid:96) (cid:18) Cξ (cid:96) (cid:96) (cid:19) (cid:96) (cid:32) C(cid:96) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ ≤ (cid:88) (cid:96) ≤ M / / log M (cid:18) Cξ √ (cid:96) (cid:19) (cid:96) · (cid:96) exp (cid:18) Cξ(cid:96) e (cid:19) · (cid:88) (cid:96) ≥ (cid:18) (cid:96) ≥ log M (cid:19) (cid:96) ≥ + (cid:88) M / / log M<(cid:96) ≤ n (cid:18) Cξ √ (cid:96) (cid:19) (cid:96) · (cid:96) exp (cid:18) Cξ(cid:96) e (cid:19) · n exp (cid:32) C(cid:96) / e √ M (cid:33) ≤ O (cid:18) M (cid:19) (cid:88) (cid:96) ≤ M / / log M (cid:96) (cid:18) Cξ √ (cid:96) e Cξ/e (cid:19) (cid:96) + (cid:88) M / / log M<(cid:96) ≤ n (cid:96) n (cid:18) Cξ √ (cid:96) e Cξ/e + C √ (cid:96) /M (cid:19) (cid:96) = o (1) , as n = o ( √ M ).Similarly, we have (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:96) <(cid:96) (cid:18) Cξ √ (cid:96) + (cid:96) (cid:96) (cid:19) (cid:96) (cid:18) Cξ (cid:96) + (cid:96) (cid:96) (cid:19) (cid:96) (cid:32) C ( (cid:96) + (cid:96) ) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ ≤ (cid:88) (cid:96) ,(cid:96) ,(cid:96) ≥ (cid:96) <(cid:96) (cid:18) Cξ √ (cid:96) (cid:96) (cid:19) (cid:96) ( Cξ ) (cid:96) (cid:32) C(cid:96) / (cid:96) ≥ √ M (cid:33) (cid:96) ≥ ≤ (cid:88) (cid:96) ≤ M / / log M (cid:96) ( Cξe
Cξ/ √ (cid:96) ) (cid:96) (cid:88) (cid:96) ≥ (cid:18) (cid:96) ≥ log M (cid:19) (cid:96) ≥ + (cid:88) M / / log M<(cid:96) ≤ n (cid:96) n (cid:0) Cξe
Cξ/ √ (cid:96) + C √ (cid:96) /M (cid:1) (cid:96) = o (1) , as n = o ( M ). By Markov’s inequality, a.a.s. there are no sets S ⊆ [ n ] where d ( S ) ≤ . | S | , d ( S ) ≤ M/ e ( S, S ) = 0. This, together with Lemma 18, completes the proof for Theorem 10(a).
We start by some structural result involving vertices in H .16 emma 19. Suppose d is a degree sequence satisfying either of the following two conditions. • |H| = ω (1) . • H = O (1) and d ( H ) ≥ M / .A.a.s. all vertices in H are contained in the same component of G ( n, d ) .Remark. This lemma does not assume M − d ( H ) = Ω( M ). It may be useful for studyingthe connectivity of G ( n, d ) where d ( H ) ≥ (1 / − o (1)) M . However we do not attempt that in thispaper. Proof of Lemma 19.
The case |H| ≥ log M follows by [16, Lemma 28]. Assume |H| < log M and |H| = ω (1). Our proof considers two cases depending on if d ( H ) is at most M/
16. When d ( H ) ≤ M/
16 we can apply the following claim.
Claim 20.
Suppose M − d ( H ) = Ω( M ) . Let u, v ∈ H . Then, for any two disjoint graphs H and H on H , and uv / ∈ H ∪ H , P (cid:0) uv ∈ G ( n, d ) | H +1 , H − (cid:1) = Ω (cid:18) M (cid:19) . Let G ( H , p ) be the binomial random graph on vertex set H where each pair of vertices areadjacent independently with probability p , and let G [ H ] be the subgraph of G ( n, d ) induced by H .By Claim 20, we can couple G [ H ] with G ( H , c/ log M ) such that G [ H ] contains G ( H , c/ log M ) as asubgraph for some constant c >
0. As |H| = ω (1) and |H| < log M , we know a.a.s. G ( H , c/ log M )is connected (the connectivity threshold for G ( H , p ) is at p = ln( |H| ) / |H| ). Hence, a.a.s. G [ H ] isconnected.Next consider the case where |H| = ω (1) and d ( H ) > M/
16. Let H + = { i : d i ≥ M / / log M } .Obviously H + is nonempty as |H| < log M and d ( H ) > M/
16 but all vertices in
H \ H + havedegree less than M / / log M . The assertion of the lemma in this case follows from the followingclaim. Claim 21.
Suppose H + (cid:54) = ∅ . Then a.a.s. v is adjacent to every vertex in H , for every v ∈ H + . Finally, if |H| = O (1) and d ( H ) ≥ M / then H + is nonempty and thus our assertion in thiscase also follows by Claim 21. This completes the proof of Lemma 19.It only remains to prove the two claims above. Proof of Claim 20.
Let W be the set of graphs G in G ( n, d ) where H ⊆ G , H ∩ G = ∅ and uv / ∈ G . Let W (cid:48) be the set of graphs G on G ( n, d ) where H ⊆ G , H ∩ G = ∅ and uv ∈ G . Then, P (cid:0) uv ∈ G [ H ] | H +1 , H − (cid:1) = | W (cid:48) || W | + | W (cid:48) | . We estimate the ratio | W (cid:48) | / | W | using switchings as follows. Given G ∈ W , a forward switchingidentifies an ordered 4-tuple of vertices ( x , x , x , x ), all of which from [ n ] \ H satisfying thefollowing conditions: (a), all six vertices u , v and x i , 1 ≤ i ≤ x = x is permitted; (b), ux , x x and vx are edges in G ; (c) x x and x x are non-edges in G . Theswitching then replaces ux , vx and x x by uv , x x and x x . The resulting graph G (cid:48) is in W (cid:48) .Given G , there are at least d u − |H| and d v − |H| ways to choose x , and x respectively. Then,there are at least M − d ( H ) − √ M / log M ) ways to choose x and x so that both vertices are17n [ n ] \ H and none of x x and x x are edges in G . Thus, the total number of forward switchingsthat can be applied to G is at least( d u − |H| )( d v − |H| )( M − d ( H ) − M/ log M ) = Ω( M / log M ) , as d u , d v ≥ √ M / log M , |H| < log M and M − d ( H ) = Ω( M ). On the other hand, the number ofways to perform a backward switching to any G (cid:48) ∈ W (cid:48) is at most M (at most M ways to choose x and x and then at most M ways to choose x and x ). Thus, | W (cid:48) | / | W | ≥ Ω( M / log M ) /M =Ω(1 / log M ). It follows now that P (cid:0) uv ∈ G [ H ] | H +1 , H − (cid:1) = Ω(1 / log M ). Proof of Claim 21.
Let v ∈ H + and u ∈ H . We will show that P ( uv / ∈ G ( n, d )) = o (log − M )and our assertion of the claim follows by taking the union bound over all pairs of ( v, u ) where v ∈ H + and u ∈ H .Let W be the set of graphs in G ( n, d ) where u is not adjacent to v and W (cid:48) = G ( n, d ) \ W .Given G ∈ W , a forward switching on G specifies a pair of vertices ( x, y ) satisfying the followingconditions: (a) x and y are both in [ n ] \ H ; (b) ux and vx are edges in G ; (c) xy is not an edgein G . The switching replaces edges ux and vy by uv and xy . The resulting graph G (cid:48) is in W (cid:48) .Given G , there are at least d u − |H| and d v − |H| ways to choose vertices x and y respectively.Among such choices, at most d u √ M / log M are such that xy is an edge (if xy is an edge then uxy is a 2-path and there are at most d u √ M / log M such 2-paths where x ∈ [ n ] \ H ). Thus, the numberof forward switchings that can be applied to G is at least( d u − |H )( d v − |H ) − d u √ M / log M = (1 + o (1)) d u d v − d u √ M / log M = (1 + o (1)) d u d v , as d v ≥ M / / log M , d u ≥ √ M / log M and |H| < log M .On the other hand, the number of ways a graph G (cid:48) can be created by a forward switching(that is, the number of backward switchings that can be applied to G (cid:48) ) is at most M . Thus, | W | / | W (cid:48) | ≤ (1 + o (1)) M/d u d v , and therefore, P ( uv / ∈ G ( n, d )) = | W || W | + | W (cid:48) | = (1 + o (1)) M/d u d v = O (log M/M / ) , as d v ≥ M / / log M and d u ≥ √ M / log M . This completes the proof of the claim.Now we are ready to prove the second main theorem about the connectivity of G ( n, d ). Proof of Theorem 12.
Expose the set of edges incident with H and let d (cid:48) = ( d i ) i ∈ [ n ] \H denotethe remaining degree sequence for vertices in [ n ] \ H . That is, d (cid:48) i = d i − x i where x i is the numberof edges between i and H . In this proof we will focus on the subgraph of G ( n, d ) induced by [ n ] \ H .Conditioning on d (cid:48) , this subgraph is distributed as G ([ n ] \ H , d (cid:48) ), a uniformly random graph on[ n ] \ H with degree sequence d (cid:48) . Note that some vertices in [ n ] \ H may have degree 0 with respectto d (cid:48) . They are not of interest for study as they are just isolated vertices in G ([ n ] \ H , d (cid:48) ), andthey are known to be adjacent to some vertex in H . Hence, let V (cid:48) be the set of vertices v ∈ [ n ] \ H where d (cid:48) v ≥
1. Let M (cid:48) = (cid:80) i ∈ V (cid:48) d (cid:48) i . Conditioning on V (cid:48) and d (cid:48) , the subgraph of G ( n, d ) induced by V (cid:48) is distributed as G ( V (cid:48) , d (cid:48) | V (cid:48) ). By the theorem hypothesis that M − d ( H ) = Ω( M ), it followsthat M (cid:48) ≥ M − d ( H ) = Ω( M ). Hence, by the definition of H we have d (cid:48) i ≤ √ M / log M = O ( √ M (cid:48) / log M (cid:48) ) , for all i ∈ V (cid:48) . (6) Claim 22.
A.a.s. there exists v ∈ [ n ] \ H such that d v ≥ and v is adjacent to H . S ⊆ [ n ], we say v is adjacent to S if there exists u ∈ S which is adjacent to v . Next, wecolour the vertices in [ n ] \ H that are adjacent to H as follows. If |H| = ω (1), or if |H| = O (1) and d ( H ) ≥ M / then let U be the set of vertices v ∈ [ n ] \ H where v is adjacent to some vertex in H .Colour all vertices in U with colour 1 and let V = U ∩ V (cid:48) . I.e. V is the subset of vertices in V withdegree at least 1 with respect to d (cid:48) . If |H| = O (1) and d ( H ) < M / , then the subgraph H inducedby the set of edges incident with H has O (1) components. Let C , . . . , C k be an enumeration ofthese components. Colour all vertices in V ( C ) \ H with colour i and let V i = V ( C ) ∩ V (cid:48) for i ∈ [ k ].I.e. V i is the set of vertices in V ( C ) \ H with d (cid:48) v ≥
1. Combining both cases and by Claim 22, wehave some 1 ≤ k = O (1) where V (cid:48) is partitioned to at most k + 1 parts. The vertices in the first k parts V , . . . , V k are coloured with colours 1,2,. . . , k respectively, and the vertices in the last partare uncoloured. By Lemma 19, we may assume that all monochromatic vertices are contained inthe same component of G ( n, d ). Suppose V i (cid:54) = ∅ for every i ∈ [ k ]. Then, the connectivity of G ( n, d )is implied if we can prove that there is no partition of V (cid:48) into S and T where e ( S, T ) = 0, and nocolour i such that both S ∩ V i (cid:54) = ∅ and T ∩ V i (cid:54) = ∅ . However this implication is not true if thereexists i where V i = ∅ , as the set of vertices in [ n ] \ H coloured i (they are all isolated vertices withrespect to d (cid:48) since V i = ∅ ) together with their neighbours in H may lie in a distinct componentfrom the vertices of other colours, and the uncoloured vertices. The next claim excludes such apossibility. Claim 23.
A.a.s. for every i ∈ [ k ] , if there is some vertex u coloured i and d (cid:48) u = 0 then V i (cid:54) = ∅ . Therefore we may assume that V i (cid:54) = ∅ for all i ∈ [ k ]. In the rest of the proof, we will focuson G ( V (cid:48) , d (cid:48) | V (cid:48) ), and we call d (cid:48) v the degree of v for v ∈ V (cid:48) . When we use graph notation such as e ( U, V ) and d ( U ), the graph referred to is G ( V (cid:48) , d (cid:48) | V (cid:48) ) unless otherwise specified. By construction,all vertices in V (cid:48) has degree at least 1. Moreover, by the theorem hypothesis on n and n and bythe facts that n (cid:48) ≤ n , n (cid:48) ≤ n and M (cid:48) = Ω( M ), where n (cid:48) is the number of uncoloured vertices ofdegree 1 in V (cid:48) , and n (cid:48) is the number of uncoloured vertices of degree 2 in V (cid:48) , it follows that n (cid:48) = o ( √ M (cid:48) ) , n (cid:48) = o ( M (cid:48) ) . (7)Now, as argued above, the connectivity of G ( n, d ) immediately follows from the following twoclaims. Claim 24.
A.a.s. there is no S ⊂ V (cid:48) \ ( ∪ i ∈ [ k ] V i ) where d ( S ) ≤ M (cid:48) / and e ( S, V (cid:48) \ S ) = 0 . Claim 25.
A.a.s. for every I ⊆ [ k ] , there exists no T ⊆ [ n ] \H where ∪ i ∈ I V i ⊆ T , T ∩ ( ∪ i ∈ [ k ] \ I V i ) = ∅ , d ( T ) ≤ M (cid:48) / and e ( T, V (cid:48) \ T ) = 0 . Now Theorem 12 follows.The proofs of Claims 24 and 25 are analogous to the proof of Theorem 10. We briefly sketchthe arguments.
Proof of Claim 24.
By (7), n (cid:48) ≤ ξ √ M (cid:48) , and n (cid:48) ≤ ξM (cid:48) , for some ξ = o (1). Moreover, by (6),joint probabilities in (1) can be applied to G ( V (cid:48) , d (cid:48) | V (cid:48) ). The rest of the proof is identical to that ofLemma 18 and Theorem 10, noting that S contains only uncoloured vertices. Proof of Claim 25.
We fix I , which fixes V I := ∪ i ∈ I V i . Let D denote the total degree of V I , i.e. D = (cid:80) u ∈ V I d (cid:48) u . Next, given a vector (cid:96) , (cid:96) , . . . , the number of ways to choose T where ∪ i ∈ I V i ⊆ T , T ∩ ( ∪ i ∈ [ k ] \ I V i ) = ∅ , and there are (cid:96) i uncoloured vertices of degree i in T is at most (cid:18) ξ √ M (cid:48) (cid:96) (cid:19)(cid:18) ξM (cid:48) (cid:96) (cid:19)(cid:18) n (cid:48) (cid:96) ≥ (cid:19) , (cid:96) ≥ = (cid:80) i ≥ (cid:96) i . By Lemma 18 we may assume that d ( T ) = D + (cid:88) i ≥ i(cid:96) i < . (cid:88) i ≥ (cid:96) i + D . The probability that e ( T, V (cid:48) \ T ) = 0 is at most (cid:18) d ( T ) M (cid:48) − d ( T ) (cid:19) d ( T ) / (cid:18) − d ( T ) M (cid:48) (cid:19) ( M (cid:48) − d ( T )) / = (cid:32) D + (cid:80) i ≥ i(cid:96) i M (cid:48) − ( D + (cid:80) i ≥ i(cid:96) i ) (cid:33) ( D + (cid:80) i ≥ i(cid:96) i ) / (cid:18) − D + (cid:80) i ≥ i(cid:96) i M (cid:48) (cid:19) ( M (cid:48) − ( D + (cid:80) i ≥ i(cid:96) i )) / . Given (cid:96) , (cid:96) , . . . , the above function is monotonely decreasing on D on the domain where ( D + (cid:80) i ≥ i(cid:96) i ) /M (cid:48) ≤ /
2. Hence, the above probability is maximised at D = 0. Hence, the probabilityof existing such a set T , given I , and (cid:96) , (cid:96) , . . . is at most (cid:18) ξ √ M (cid:48) (cid:96) (cid:19)(cid:18) ξM (cid:48) (cid:96) (cid:19)(cid:18) n (cid:48) (cid:96) ≥ (cid:19) (cid:32) (cid:80) i ≥ i(cid:96) i M (cid:48) − (cid:80) i ≥ i(cid:96) i (cid:33) (cid:80) i ≥ i(cid:96) i / (cid:18) − (cid:80) i ≥ i(cid:96) i M (cid:48) (cid:19) ( M (cid:48) − (cid:80) i ≥ i(cid:96) i ) / ≤ (cid:18) ξ √ M (cid:48) (cid:96) (cid:19)(cid:18) ξM (cid:48) (cid:96) (cid:19)(cid:18) n (cid:48) (cid:96) ≥ (cid:19) (cid:18) (cid:80) i ≥ i(cid:96) i M (cid:48) (cid:19) (cid:80) i ≥ i(cid:96) i / where (cid:80) i ≥ i(cid:96) i < . (cid:80) i ≥ (cid:96) i implying (cid:96) ≥ ≤ (cid:96) + (cid:96) , (cid:96) + 2 (cid:96) + 3 (cid:96) ≥ (cid:96) ≤ . (cid:96) + (cid:96) + (cid:96) ≥ ) ≤ (cid:96) + 5 (cid:96) . The rest of the analysis is the same as in Theorem 10.Now we prove Claims 22 and 23. Both claims concern events related to edges incident withvertices in H . We will use switchings to bound probabilities of such events. Proof of Claim 22.
Let E denote the event that e ( H , [ n ] \ H ) > H and [ n ] \ H has one end whose degree equals 1 in G ( n, d ). If there is no vertex v ∈ [ n ] \ H where d v ≥ v is adjacent to H , then we must have either E or e ( H , [ n ] \ H ) = 0. It is then sufficientto show that P ( E ) = o (1) and P ( e ( H , [ n ] \ H ) = 0) = o (1). Let G be the class of graphs in G ( n, d )where e ( H , [ n ] \ H ) = 0 and let G (cid:48) be the class of graphs in G ( n, d ) where e ( H , [ n ] \ H ) = 2. If G = ∅ then P ( e ( H , [ n ] \ H ) = 0) = 0. Otherwise, P ( e ( H , [ n ] \ H ) = 0) ≤ |G||G (cid:48) | . We define a switching from G ∈ G by choosing an edge xy in G [ H ] and another edge uv in G [[ n ] \H ] .Replace these two edges by xu and yv . The resulting graph G (cid:48) is in G (cid:48) . There are d ( H ) / √ M / log M ) ways to choose xy and d ([ n ] \ H ) = Ω( M ) ways to choose uv . So the number ofswitchings applicable on G is at least Ω( M / / log M ). On the other hand, for every G (cid:48) , it can beproduced by at most 1 way, as there are exactly 2 edges between H and [ n ] \ H . It follows thenthat P ( e ( H , [ n ] \ H ) = 0) ≤ |G||G (cid:48) | = O (log M/M / ) , as desired. 20ext, let G be the class of graphs in G ( n, d ) ∩ E , and let G (cid:48) be the class of graphs in G ( n, d )where there is exactly one neighbour of H in [ n ] \ H with degree at least 2, and all the otherneighbours of H in [ n ] \ H have degree equal to 1. Define a switching from G to G (cid:48) as follows.Given G ∈ G , choose 4 vertices ( u, v, x, y ) such that u ∈ H , v, x, y ∈ [ n ] \ H , uv and xy are edges,and d x ≥
2. Since d x > d v = 1 it follows immediately that ux and vy are not edges. Theswitching replaces uv and xy by ux and vy . The resulting graph G (cid:48) is in G (cid:48) since x becomes aneighbour of H and d x ≥
2. There is at least one way to choose v , since G ∈ E . The total degree of[ n ] \ H is Ω( M ) by the theorem assumption, and there are at most o ( √ M ) vertices in the set whosedegree is 1. Moreover, all vertices in [ n ] \ H has degree at most √ M / log M . Hence, there are atleast Ω( M ) / ( √ M / log M ) = Ω( √ M log M ) vertices in [ n ] \ H whose degree is at least 2. Hence,the number of choices for x and y is Ω( √ M log M ). Thus, the number of switchings that can beapplied to G is Ω( √ M log M ). On the other hand, given G (cid:48) ∈ G , there is a unique way to choose u and x , and at most n ways to choose v and y so that an inverse switching can be applied. Thus, P ( E ) = O (cid:18) n √ M log M (cid:19) = o (1) , and our assertion follows. Proof of Claim 23.
We consider two cases. In the first case, we assume |H| = O (1) and d ( H ) ≤ M / . We prove that a.a.s. for every vertex u ∈ H , u is adjacent to some vertex v ∈ V (cid:48) .That will imply the assertion in the claim. Fix u ∈ H . Let G be the set of graphs in G ( n, d ) wherefor each neighbour v ∈ [ n ] \ H of u , d (cid:48) v = 0. Let G (cid:48) be the set of graphs in G ( n, d ) where for allbut exactly one neighbours v ∈ [ n ] \ H of u , d (cid:48) v = 0. Define a switching from G to G (cid:48) as follows.Let G ∈ G . Choose a neighbour v ∈ [ n ] \ H of u , and then choose two vertices ( u (cid:48) , v (cid:48) ) in [ n ] \ H such that d (cid:48) v (cid:48) ≥ u (cid:48) v (cid:48) is an edge. Since d (cid:48) v = 0 and d (cid:48) v (cid:48) > u (cid:48) v and uv (cid:48) are not edges. Then the switching replaces edges uv and u (cid:48) v (cid:48) by uv (cid:48) and u (cid:48) v . The resulting graph G (cid:48) is in G (cid:48) because d (cid:48) v (cid:48) ≥ G (cid:48) and v (cid:48) is adjacent to u . Given G , the number of choices for v is at least √ M / log M since d u ≥ √ M / log M . By definition of G , we know (cid:80) x d (cid:48) x = 0 wherethe summation is over all x ∈ [ n ] \ H that are neighbours of u . However, since d ( H ) < M / , M (cid:48) = (cid:80) y ∈ [ n ] \H d (cid:48) y ≥ M − M / . Let n (cid:48)(cid:48) be the number of vertices in V (cid:48) with degree (with respectto d (cid:48) ) one. Since every edge incident with H can create at most one new vertex v with d (cid:48) v = 1, wemust have n (cid:48) ≤ n + d ( H ) < M / . It follows that (cid:88) y ∈ [ n ] \H d (cid:48) y ≥ ≥ ( M − M / ) − n (cid:48) ≥ (1 − o (1)) M. Since d (cid:48) y ≤ d y ≤ √ M / log M for every y ∈ [ n ] \ H , the number of choices for v (cid:48) is at least((1 − o (1)) M ) / ( √ M / log M ) = Ω( √ M log M ). Consequently, the number of ways to perform aswitching to G ∈ G is at least ( √ M / log M ) · Ω( √ M log M ) = Ω( M ). On the other hand, for every G (cid:48) ∈ G , G (cid:48) can be created by at most O ( M / ) different switchings, since given G (cid:48) there is at mostone way to choose v (cid:48) , and at most n + M / ≤ M / ways to choose v , who must satisfy d (cid:48) v = 1.Hence, the probability that u is not adjacent to any vertex in V (cid:48) is at most |G||G (cid:48) | = O ( M / /M ) = O ( M − / ) . Our claim in this case follows by taking the union bound over the O (1) vertices u ∈ H .In the second case, we assume H = ω (1) or d ( H ) ≥ M / . In this case, all vertices adjacent to H are coloured with 1. Let P denote the set of graphs in G ( n, d ) with the property in Claim 22,21.e. there exists v ∈ [ n ] \ H where d v ≥ v is adjacent to H . Let G denote the set of graphsin P where for all v ∈ [ n ] \ H adjacent to H , d (cid:48) u = 0. Let G (cid:48) be the set of graphs in P wherethere are exactly two vertices v , v ∈ [ n ] \ H (cid:48) such that d (cid:48) v = 1, d (cid:48) v ≥
1, and d (cid:48) v = 0 for all v ∈ ([ n ] \ H ) \ { v , v } . Define a switching from G to G (cid:48) as follows. Given G ∈ G , the switchingchooses 4 vertices ( u, v, x, y ) such that u ∈ H , v, x, y ∈ [ n ] \ H , uv and xy are edges, d v ≥ d (cid:48) x ≥
2. Since d (cid:48) x > d (cid:48) v = 0, we know that ux and vy are not edges. The switching replacesedges uv and xy by ux and vy . Let G (cid:48) denote the resulting graph. In G (cid:48) , both x and v are adjacentto H , since x is adjacent to u ∈ H , and v is adjacent to some vertex u (cid:48) ∈ H where u (cid:48) (cid:54) = u , as d v ≥ d (cid:48) v was equal to 0 in G . Let U be the set of uncoloured vertices. The total degree of U is at least M (cid:48) = Ω( M ), since e ( U, [ n ] \ U ) = 0 in G by the definition of G . Moreover, there areat most o ( √ M ) vertices in U of degree one by (7). Thus, the number of choices for x and y is atleast Ω( M ) − o ( √ M ) = Ω( M ). Consequently, the total number of ways to perform a switchingon G is at least Ω( M ). On the other hand, given G (cid:48) ∈ G (cid:48) , there are exactly 2 vertices in [ n ] \ H whose degrees (with respect to d (cid:48) ) are at least 1, and they must be v (the one with degree equalto 1) and x . Fixing v fixes y as d (cid:48) v = 1. Given x there are at most √ M / log M ways to choose u as d x ≤ √ M / log M . Hence, the number of ways G (cid:48) can be created via a switching is at most2 √ M / log M . Together with Claim 22, P ( V = ∅ ) ≤ |G||G (cid:48) | + P ( ¬P ) ≤ √ M / log M Ω( M ) + o (1) = o (1) . This completes the proof of the claim.
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