On the chromatic number of random graphs with a fixed degree sequence

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1 On the chromatic number of random graphs with a fixed degree sequence Alan Frieze Michael Krivelevich Cliff Smyth December 13, 006 Abstract Let d = 1 d 1 d d n be a non-decreasing sequence of n positive integers, whose sum is even Let G n,d denote the set of graphs with vertex set [n] = {1,,,n} in which the degree of vertex i is d i Let G n,d be chosen uniformly at random from G n,d Let d = d 1 + d + + d n )/n be the average degree We give a condition on d under which we can show that whp the chromatic number of G n,d is Θd/ ln d) This condition is satisfied by graphs with exponential tails as well those with power law tails 1 Introduction Let d = 1 d 1 d d n be a fixed non-decreasing sequence of n positive integers, whose sum is even Let G n,d denote the set of graphs with vertex set [n] = {1,,,n} in which the degree of vertex i is d i Let G n,d be chosen uniformly at random from G n,d When d i = r for i [n] then this models a random r-regular graph G n,r and there is a large literature on this subject We refer the reader to the survey by Wormald [18] for an excellent summary By now we know much about the structure of random regular graphs For general d, less is known In many, but not all, cases we can estimate G n,d See Bender and Canfield [5], McKay and Wormald [13, 14] We have the configuration model to study them, Bollobás [6] We know something of their connectivity properties, Molloy and Reed Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh PA1513, USA Supported in part by NSF grant CCR Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 69978, Israel krivelev@posttauacil Research supported in part by USA-Israel BSF Grant and by grants 64/01 and 56/05 from the Israel Science Foundation Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 0139, USA 1

2 [16, 17]See also Cooper and Frieze [8] for the connectivity properties of random digraphs with a fixed degree sequence) They have been used in the context of massive graph models of telephone networks and the WWW, Aiello, Chung and Lu [3] In this paper we will concern ourselves with the chromatic number of G n,d What should we expect? For r-regular graphs it is known that when r is large χg n,r ) r ln r whp1, see Frieze and Luczak [10] and Achlioptas and Moore [1] see also Cooper et al [9] for extension of some of these results to the case where r = rn) n 1 η, for an arbitrary small constant η > 0) For the random graph G n,p where the average degree d = n 1)p is large, it is known that χg n,p ) d, see Bollobás [7], Luczak [1] and Achlioptas and Naor [] ln d So if we let d = d 1 + d + + d n n be the average degree in G n,d then we might hope to prove that χg n,d ) d whp This ln d is too much to expect given the variety of possible degree sequences Indeed if ν = n d i = { 1 i n ν d ln n i > n ν ln n and then the average degree d, but whp the sub-graph H induced by the ν largest degree vertices has average degree close to d ln n when d is large Then whp χh) lnd = oln lnn) and so we will have to be less ambitious in our goals Let be the sum of the k largest degrees D k = d n + d n d n k+1 dln n for ln ln n Let Theorem 1 M 1 = D n = dn and M = n d i d i 1) M 1 where = d n i=1 1 Suppose that there exist constants 1/ < α < 1, ǫ, K 0 > 0 and ω = ωn) such that a) for k ǫn b) 5 M /ω D k K 0 dnk/n) α 1) 1 A sequence of events E n occurs with high probability whp) if lim n PrE n ) = 1 The methods of [9] can be used to verify this claim

3 Then there exists b 1 dependent only on α,ǫ,k 0 such that whp χg n,d ) b 1 d ln d Suppose only that 4 M 1 /ω a weaker condition than 1b)), then there exists b such that whp χg n,d ) b d ln d Notice that if Condition a) of the theorem holds for α,ǫ,k 0 then it also holds when K 0 is replaced by a larger constant As we increase K 0 our bound b 1 will decrease We are therefore justified in assuming throughout that K 0 is sufficiently large that some inequalities are valid Condition b) is chosen so that we can use the results of [15] It may be possible to prove our results under the less stringent conditions of [14], but there are difficulties, as will be pointed to later A referee has suggested that the quantity d = max k Dk /dnk which can be viewed as the probabilistic version of the maximum average degree ) might be a better predictor of chromatic number This could well be true and it makes for an interesting research question We also suspect that if d i dfi/n) where f is some nice real valued function on [0, 1] such that 1 fx)dx = 1 then we might be able to determine the chromatic number 0 of G n,d asymptotically The proof of the upper bound is given in Section 3 The proof of the lower bound is given in Section 4 We have made no attempt to optimize constants Degree sequences that satisfy 1) It is natural to ask whether there many types of degree sequence that satisfy the conditions of the first part of the theorem We first consider a degree sequence with a power law tail For integer l 1 we let ν l denote the number of vertices of degree l Our assumption is that there are some constants A > 0 and γ > 3 such that for l A/ǫ) 1/γ 1) Here we have α = γ γ 1 > 1/ 0 l 1 ν l Adl γ n l n 1/5 / lnn 0 l > n 1/5 / ln n 3

4 We have forced Condition b) and so we only need to check Condition a) Suppose now that k = κn ǫn Then where δ r 0 as r and { r = min D k Adn l r ρ : Ad l ρ l γ 1) = 1 + δ r ) Adn γ r γ l γ κ } ) 1/γ 1) Ad = 1 + δ κ) γ 1)κ where δ κ 0 as κ 0 Thus, ) α γ 1 D k 1 + δ r )1 + δ κ) γ γ ) 1 A dnκ α Ad and we have the condition of Theorem 1 A similar argument holds if we assume that our degree sequence has an exponential tail viz for some constants A > 0 and 0 < ǫ γ < 1 we have for l ln 1/γ Ad/ǫ) ν l Adγ l n Note that whp the degree sequence of G n,p, p = c/n, c constant, satisfies this condition Suppose now that k = κn ǫn Then where as ǫ 0 Thus, D k Adn l r lγ l Adn x=r 1 xγ x dx = Adnγr 1 ln 1/γ { r = min ρ : Ad } ) Ad lγ l κ ln 1/γ κ ln 1/γ l ρ D k κn and we have the condition of Theorem 1 with room to spare r ) ln 1/γ 3 Upper bound 31 Configurations We will work initially in the configuration model and then show how our results can be justified in the uniform model Let W = [nd] be our set of points and let W i = [d 1 + 4

5 + d i 1 + 1,d d i ], i [n], partition W The function φ : W [n] is defined by w W φw) Given a pairing F ie a partition of W into m = dn/ pairs) we obtain a multi-)graph G F with vertex set [n] and an edge φu),φv)) for each {u,v} F Choosing a pairing F uniformly at random from among all possible pairings of the points of W produces a random multi-)graph G F This model is valuable because of the following easily proven fact: Suppose G G n,d Then PrG F = G G F is simple) = 1 G n,d It follows that if G is chosen randomly from G n,d, then for any graph property P PrG P) PrG F P) PrG F is simple) ) Before starting the proof of the upper bound in Theorem 1, we introduce some notation that we will use throughout the course of the proof Let { } θ = max d /α+1) 16, α 1 and for some sufficiently large K 1 { ) } /α 1) α 1)θ ǫ 1 = min ǫ, 3) K 1 d For a set U V G F ) we denote by DU) the sum of the degrees of U in G F We first prove Lemma 1 Whp G F does not contain a set of vertices S with k = S ǫ 1 n, such that S induces a sub-graph G[S] with minimum degree at least θ Proof Suppose that there exists a vertex set S of size k [θ,ǫ 1 n] that induces a graph G[S] with minimum degree at least θ Let S 1 be the k/ lowest degree vertices of S and let S = S \ S 1 For vertex sets S,T let es) denote the number of edges contained in S and let es : T) denote the number of edges joining S and T At least one of the following two events must occur: E 1 : es 1 : S ) θk/4; E : es 1 ) θk/8 5

6 This follows because es 1 ) + es 1 : S ) = v S 1 d G[S] v) θk/ Now k ǫ 1 n and so, provided dn θk/ dn θǫ 1 n/ dn ) α 1/ 1600 K 1 16 α 1 We have chosen K 1 large enough so that whichever of d /α+1), 16 α 1 is the larger, θǫ 1 < 1/100) And so PrE 1 ) ǫ 1 n S =k ǫ 1 n ǫ 1 n ǫ 1 n n k en k en k ǫ 1 n k ) ) θk/4 DS1 ) DS ) 4) θk/4 dn θk/ ) Dk/ θk/4 ) Dk/ ) k 5eD k/ θkdn ) k 5eK0d α θ dn θk/ ) θk/4 ) θk/4 ) ) α 1 θk/4 k n ) α 1)θ/4 1 ) ) 5e K0d θ/4 k n α θ ǫ 1 n k ) 3α 1)θ/16 ) ) 5e K0d θ/4 k n α θ ǫ 1 n k ) θk/4 = o1), n ) α 1)/4 5e K 0 α if we choose K 1 100e 4 K 4 0 we used the assumption k/n ǫ 1 α 1)θ/K 1 d)) /α 1) 16/K 1 ) /α 1) in the last line) Explanation of 4): At least θk/4 of the points corresponding to vertices of S 1 are to be paired with points corresponding to vertices of S When pairing points, the probability of pairing with a point corresponding to S is never more than DS ), given up to l/ 1 dn l previous pairings 6

7 In the following estimate, l = ls 1 ) is the total degree of the subgraph G[S 1 ] Here θk/4 l K 0 ǫ α 1dn Now the maximum degree in S 1 is at most D k/ K k/ 0dn/k) 1 α and so PrE ) ǫ 1 n K 0 ǫ α 1 dn S 1 ={i 1 < <i k/ } l=θk/4 a 1 + +a k/ =l k/ ) l/ ) dn l) l/ k/ a 1,a,a k/ is the degree sequence of G[S 1 ] more explanation below) t=1 d a t i t 5) ǫ 1 n K 0 ǫ α 1 dn S 1 ={i 1 < <i k/ } l=θk/4 a 1 + +a k/ =l ǫ 1 n K 0 ǫ α 1 dn S 1 ={i 1 < <i k/ } l=θk/4 a 1 + +a k/ =l ǫ 1 n K 0 ǫ α 1 dn S 1 ={i 1 < <i k/ } l=θk/4 ǫ 1 n ) n 64K dn 0 dk α 1 k/ θn α 1 ǫ 1 n en 64K dn 0 dk α 1 k θn α 1 ǫ 1 n 1 K0 4 dn d k α 1 θ n α 1 = o1) k 64K 0 dk α 1 ) θk/16 θn α 1 ) θk/8 ) θ/4 ) k/ ldn ) l/ k/ K 0 dn/k) 1 α) ) a t t=1 16K 0 dk α 1 θn α 1 ) l/ Here we choose K 1 10 K 0 so that 1 K 4 0d ǫ α 1 1 < θ / Explanation of 5): Fix S 1 = {i 1 < < i k/ } Now fix the degrees a 1,,a k/ of i 1,,i k/ k/ )) l/ is a crude upper bound on the number of graphs with this set of degrees Then we use the fact that the probability an edge exists between two vertices of degrees d r,d s is never more than d rd s, given up to l/ 1 previous pairings dn l ) l/ Now let and I t = t 1 = 1 + log 3 /ǫ 1 ) { n n 3 + 1,,n n } t 3 t+1 for t = 0, 1,,t 1 7

8 Let G t be the sub-graph of G F induced by I t and for k I t let d k be the degree of k in G t Let t = where K = 3ǫ 1 α 1)t 0 + K + K where and K,K are defined below K d α 1)t t 0 = log 3 1/ǫ For 0 t t 1 let B t = {k I t : dk t }, Z t = B t and Z = Z Z t1 Lemma Z ǫ 1 n/ whp Proof If k I t, t t 0, then there are at least n t i=1 I i 3ǫn vertices of degree at least d k, and thus d k 3ǫ 1 d and so B t =, 0 t t 0 For t > t 0 and k I t we see that d ) k is stochastically dominated by the binomial B d k, D n/3 t Furthermore, since dn k n ǫn we have and So, if K = K 0 e 3 1 α then d k D n/3 t+1 n/3 t+1 K 03 t+1)1 α) d 6) ) Pr dk t We used: Pr[Bn,p) k] n k Thus, Now, D n/3 t dn K 03 αt K α e K 3/) α 1)t ) p k ) enp k) EZ) n k t 1 t=1 3 t e t < ne t 1 ) t e t α 1)t 1 α 1)+log 3 /ǫ)+ α 1 log 3 K 1 d α 1)θ)) +log 3 /ǫ)+ α 1 log 3 K 1/α 1)) d γ where So, if γ = 1 ln 4 α + 1 ln 3 5, ) 7) 3 K = d 1/3 +log 3 /ǫ)+ α 1 log 3 K 1/α 1)) ln 4 + ) α 1 lnk 1d/α 1)), 8

9 then EZ) < e d/3 γ /α 1))lnK 1 d/α 1)) n/4 ǫ 1 n/4 It is straightforward to show that Z is concentrated about its mean A random pairing can be obtained by choosing a random permutation ω = w 1,w,,w dn ) of W and pairing w i 1 with w i, for 1 i dn/ Interchanging two elements in ω can change Z by at most two Lemma 11 of [11] states: Let S be a set with S = N Let Ω be the set of N! permutations of S Let ω be chosen uniformly from Ω Let Z = Zω) be such that Zω) Zω ) 1 when ω is obtained from ω by interchanging two elements of the permutation Pr Z EZ) t) e t /N 8) Using 8) with N = dn and t = dn) 1/ ln n yields the lemma We now appeal to a result of Alon, Krivelevich and Sudakov [4] to show that we can colour the graphs Γ t = G t B t with few colours The main result of that paper is Theorem There exists an absolute positive constant C 0 such that the following holds Let G be a graph with maximum degree in which the neighbourhood of any vertex v spans at most /f edges Then the chromatic number of G is at most C 0 ln f We use this to prove Lemma 3 There exists β > 0 such that Pr t [0,t 1 ] : χγ t ) C ) 0 t = o1) β ln d Proof Fix t and v I t and condition on the neighbours of v in G F so that v / B t In so doing we fix at most pairs of the configuration With this conditioning the number of edges ξ v in the Γ t -neighbourhood of v is stochastically dominated by the Binomial B t/, /dn ) So, if β = 4/3 γ > 0 γ as in 7)) then ) ) ) Pr ξ v t t / t /d β ) e t /d β d β t/d β dn d 1 β n But n 1/3 and t /dβ 10 for sufficiently large K 1 and so ) Pr ξ v t = o1/n) d β Applying Theorem with = t and f = d β yields the lemma Lemma 4 If F is chosen uniformly from the set Ω of pairings then whp χg F ) b 1 d ln d 9

10 Proof Let B t,g t, Γ t,t [0,t 1 ], be as defined above It follows from Lemma 3 that whp the number of colours needed to colour t 1 t=0 Γ t is at most t 1 t=0 ) C 0 K d d α 1)t β lnd = O lnd Finally note that the number of vertices in B = t 1 t=0 B t {i n n } is whp less than 3 t 1 +1 ǫn and then, by Lemma 1, the graph induced by B is θ = d /α+1) -degenerate and so can be coloured with at most θ + 1 = Od/ ln d) colours 3 From configurations to graphs It is at this point that we appeal to some results from Mckay and Wormald [15] Where possible, we will use the terminology and notation of that paper A loop of a pairing F is a pair {u,v} such that φu) = φv) A double pair of F is a pair {u 1,v 1 },{u,v } F such that φu 1 ) = φu ) and φv 1 ) = φv ) A double loop of F is a pair of pairs {u 1,v 1 }, {u,v } such that φu 1 ) = φv 1 ) = φu ) = φv ) A triple pair is a triple of pairs {u i,v i }, i = 1,, 3 such that φu 1 ) = φu ) = φu 3 ) and φv 1 ) = φv ) = φv 3 ) In the lemmas that follow, we will assume always that Condition b) of Theorem 1 holds Lemma 5 Lemma of [15]) The probability that F contains at least one triple pair is O M /M 3 1) = o1) and the probability of at least one double loop is O M /M 1) = o1) Let now l denote the number of loops and r denote the number of double pairs in F Lemma 6 Lemma 3 of [15]) If λn) then whp l + λ and r + λ 9) We define the following two operations on a pairing: If φu) = i then we say that u is in cell i I l-switching Take pairs {p 1,p 6 }, {p,p 3 }, {p 4,p 5 } where {p,p 3 } is a loop, and p 1,,p 6 are in five different cells Replace these pairs by {p 1,p }, {p 3,p 4 }, {p 5,p 6 } In this operation, none of the pairs created or destroyed is permitted to be part of a double pair See Figure 1) 10

11 p p 3 p p 3 p 1 p 4 p 1 p 4 p 6 p 5 p 6 p 5 Figure 1: II r-switching Take pairs {p 1,p 5 }, {p,p 6 }, {p 3,p 7 },{p 4,p 8 } where φp ) = φp 3 ) and φp 6 ) = φp 7 ), but the cells containing p 1,p,p 4,p 5,p 6,p 8 are all distinct Replace these pairs by {p 1,p }, {p 3,p 4 }, {p 5,p 6 },{p 7,p 8 } In this operation, none of the pairs created or destroyed other than the pairs {p,p 6 }, {p 3,p 7 }) is permitted to be part of a multiple pair See Figure ) A forward l-switching is an l-switching as described, and a backward l-switching is the reverse operation We use the same convention for r-switchings Note that a forward l- switching always reduces the number of loops by one and does not create or destroy double pairs Similarly, a forward r-switching reduces the number of double pairs by one and neither creates nor destroys loops Now let C l,r denote the set of pairings F with l loops, r double pairs and no triple pairs or double loops Lemma 7 Lemma 4 of [15]) Denote an operation taking an element of C i,j to an element C k,l by C i,j C k,l For each of the following operations, we bound the number, m, of ways of applying the operation to a fixed F 11

12 p 1 p5 p 1 p 5 p p 6 p p 6 p 3 p 7 p 3 p 7 p 4 p 8 p 4 p 8 Figure : 1) Forward l-switching C l,r C l 1,r : lm 1 m lm 1 ) Backward l-switching C l 1,r C l,r : 1 O + l + r M 1 6l + r) + l) M 1 M m M 1 M 1 M 3) Forward r-switching C 0,r C 0,r 1 : 4rM 1 m 4rM 1 1 O + r M 1 )) ) + ) M 1 )) 4) Backward r-switching C 0,r 1 C 0,r : M m M 1 16r ) ) M Now consider the following algorithm for generating a member of G n,d : 1 Generate a random pairing F If there is a double loop or a triple pair, output construed as failure 1

13 3 If the number of loops l + log n or the number of double pairs r + log n, output construed as failure 4 F 0 F 5 For i = 1 to l choose a random forward l-switching on F i 1, creating F i C l i,r 6 For i = l + 1 to l + r choose a random forward r-switching on F i 1, creating F i C 0,r i l) 7 Output G = G Fl+r G n,d For each l,r satisfying 9), with λ = log n, and G G n,d, there are by Lemma 7),4) )) l l + r) M 1 M ) l M r 1 + O + l + r + r) M M 1 M sequences of switchings which yield G Each of these has probability )) l M1 )l l!4m1 )r r!) l + r) + O + r + r) M 1 M 1 of being followed by the algorithm, given l,r Thus if Condition b) holds, then whp the algorithm outputs a graph in G n,d and PrG = G) = 1 + o1)) and so for G 1,G G n,d +log n l=0 +log n r=0 PrG = G 1 ) = 1 + o1))prg = G ) M1M l r+l Prl loops,r double pairs) l M l+r) 1 l!r! Given this, we only have to show that whp χg ) = OχG F )) Thus let H be the graph induced by the edges that are added by the switchings We will show that whp H) 5 10) Since χg ) χh)χg F ) H) + 1)χG F ), this will complete the proof of our upper bound Now every edge at distance from the loop or double edge can be used as one of the two edges destroyed by the two types of switching Thus, vertex i has probability ) ) d i O = O M 1 M 1 13

14 of having an H-edge created in any switching, regardless of the history of the switchings to this point Thus, for some constant c > 0, assuming due to Lemma 6 that G satisfies 9): ) ) log n c Pr H) 5) n nm 5/ 15/4 1 = o1) 5 and this completes the proof of the upper bound in Theorem 1 Note that we cannot immediately use the stronger result of [13] This is because in that paper Lemma 74) is replaced by an average bound M 1 4 Lower bound We will work with the configuration model We first observe that PrG F is simple) e 11) This follows directly from the formula for the asymptotic number of labelled graphs with a degree sequence in which = om 1/4 1 ), see [13] Let d t = b lnd, where b > 0 is a sufficiently small constant Fix a partition V 1,,V t ) of the vertex set of G F into t parts We will show: PrV 1,,V t ) is a proper colouring of G F ) t n This will be enough to beat both the union bound for the number of t-partitions which is t n ) and the inverse probability that G F is simple which is e see 11)) Observe first that PrV i is independent in G F ) 1 DV ) DVi )/ i)/ dn Explanation: Match the first DV i )/ points of V i in the configuration model When matching a point, it should be matched either to a point belonging to a vertex outside of V i, or to a point of the same vertex Now, if in the partition V 1,,V t ) one of the sets V i satisfies DV i ) nd/4, by the above argument we get taking b small enough if necessary): PrV 1,,V t ) is a t-colouring) PrV i is independent) e 1 o1))nd/64 t n 14

15 Assume therefore that V i dn/4 for all 1 i t Assume wlog that DV 1 ) DV ) DV t ) Choose 1 k t so that such a k exists as DV i ) nd/4 for all i dn k 4 DV i ) dn ; i=1 We will estimate from above the probability that all of the sets V i, i = 1,,k, are independent In order to do this, we will expose the pairings of the points of V i, each time choosing a set V i that has at least half of its points unpaired, for as long as possible Let σ describe the order in which the sets V 1,,V k are exposed Observe that PrV σi) is independent V σ1),,v σi 1) are independent) 1 DV σi))/4 dn match the first half of yet unmatched points of V σi) ) ) DVσi) )/4 e DV σi) ) 4 DVσi) ) 16dn Suppose that we have in this way exposed the pairings of sets V σ1),,v σl) If l i=1 DV σi) ) dn 16, then at this moment the number of paired points does not exceed dn/8, and therefore at least one of the unexposed sets V i, i {σ1),,σl)}, has at least a half of its points unpaired, and we can proceed Thus, if by the end of the process we have exposed pairings of sets V σ1),,v σl ), it follows that l i=1 DV σi)) dn/16 and therefore Hence l i=1 DV σi) ) 4 DV σi) )) 1 o1))d n 1 o1))d n 56l 56t PrV σ1),,v σl ) are independent) { exp l i=1 DV σi)) 4 DV σi)) ) 16dn } { exp dn } 4100t Recalling the expression for t and choosing the constant b there to be small enough, we get the desired estimate This completes the proof of Theorem 1 Acknowledgement: We thank Paul Horn for pointing out an error in an earlier draft of the paper 15

16 References [1] D Achlioptas and C Moore, The Chromatic Number of Random Regular Graphs, Proceedings of RANDOM 004, 19-8 [] D Achlioptas and A Naor, The Two Possible Values of the Chromatic Number of a Random Graph, Annals of Mathematics 6 005), [3] W Aiello, F Chung and L Lu A random graph model for power law graphs, Experimental Mathematics 10, 001), [4] N Alon, M Krivelevich and B Sudakov, Coloring graphs with sparse neighborhoods, Journal of Combinatorial Theory, Series B ), 73-8 [5] EA Bender and ER Canfield, The asymptotic number of labelled graphs with given degree sequences, Journal of Combinatorial Theory, Series A ) [6] B Bollobás, A probabilistic proof of an asymptotic formula for the number of labelled regular graphs, European Journal on Combinatorics ) [7] B Bollobás, The chromatic number of random graphs, Combinatorica ) [8] C Cooper and AM Frieze, The size of the largest strongly connected component of a random digraph with a given degree sequence, Combinatorics, Probability and Computing ) [9] C Cooper, AM Frieze, BA Reed and O Riordan, Random regular graphs of nonconstant degree: independence and chromatic number, Combinatorics, Probability and Computing 11 00) [10] AM Frieze and T Luczak, On the independence and chromatic numbers of random regular graphs, Journal of Combinatorial Theory Series B ) [11] AM Frieze and BGPittel, Perfect matchings in random graphs with prescribed minimal degree, Trends in Mathematics, Birkhauser Verlag, Basel 004) [1] T Luczak, The chromatic number of random graphs, Combinatorica ) [13] BD McKay and NC Wormald, Asymptotic enumeration by degree sequence of graphs with degree on 1/ ), Combinatorica ) [14] BDMcKay and NCWormald, Asymptotic enumeration by degree sequence of graphs of high degree, European Journal of Combinatorics ) [15] BD McKay and NC Wormald, Uniform generation of random regular graphs of moderate degree, Journal of Algorithms )

17 [16] M Molloy and BA Reed, A Critical Point for Random Graphs with a Given Degree Sequence, Random Structures and Algorithms ) [17] M Molloy and BA Reed, The Size of the Largest Component of a Random Graph on a Fixed Degree Sequence, Combinatorics, Probability and Computing ) [18] NC Wormald Models of random regular graphs, Surveys in Combinatorics, London Mathematical Society Lecture Note Series 67, Cambridge University Press, Cambridge, 1999 JDLamb and DAPreece, Eds), Proceedings of the 1999 British Combinatorial Conference, Cambridge University Press,