Number of 1-factorizations of regular high-degree graphs
aa r X i v : . [ m a t h . C O ] J un Number of 1-factorizations of regular high-degree graphs
Asaf Ferber ∗ Vishesh Jain † Benny Sudakov ‡ Abstract A -factor in an n -vertex graph G is a collection of n vertex-disjoint edges and a -factorization of G isa partition of its edges into edge-disjoint -factors. Clearly, a -factorization of G cannot exist unless n iseven and G is regular (that is, all vertices are of the same degree). The problem of finding -factorizationsin graphs goes back to a paper of Kirkman in 1847 and has been extensively studied since then. Decidingwhether a graph has a -factorization is usually a very difficult question. For example, it took more than 60years and an impressive tour de force of Csaba, Kühn, Lo, Osthus and Treglown to prove an old conjectureof Dirac from the 1950s, which says that every d -regular graph on n vertices contains a -factorization,provided that n is even and d ≥ ⌈ n ⌉ − . In this paper we address the natural question of estimating F ( n, d ) , the number of -factorizations in d -regular graphs on an even number of vertices, provided that d ≥ n + εn . Improving upon a recent result of Ferber and Jain, which itself improved upon a result ofCameron from the 1970s, we show that F ( n, d ) ≥ (cid:0) (1 + o (1)) de (cid:1) nd/ , which is asymptotically best possible. A -factorization of a graph G is a collection M of edge-disjoint perfect matchings (also referred to as -factors) whose union is E ( G ) . An equivalent definition of a -factorization is an edge-coloring of G whereeach color class consists of a perfect matching. Clearly, if G admits a -factorization then the number ofvertices of G , denoted by | V ( G ) | , is even, and G is a regular graph (that is, all its vertices are of the samedegree). The problem of finding -factorizations in graphs goes back to a paper of Kirkman [17] from 1847and has been extensively studied since then in graph theory and in combinatorial designs (see, e.g., [21, 27]and the references therein). Despite the fact that -factorizations of the complete graph are quite easy toconstruct (for example, see [20]), the problems of enumerating all the distinct -factorizations and finding -factorizations in graphs which are not complete are considered as much harder.A simple example is the problem of finding (and asymptotically enumerating) -factorizations of K n,n , thecomplete bipartite graph with both parts of size n . Note that a -factorization of K n,n is equivalent to a Latinsquare, where a Latin square is an n × n array, with each row and each column being a permutation of { , . . . , n } (in particular, each element appears exactly once in each row and each column). The existence of Latin squaresfollows easily from Hall’s marriage theorem. On the other hand, in order to prove an asymptotic formula forthe number of Latin squares, one needs sophisticated estimates of the permanent of the adjacency matrix ofregular bipartite graphs. Given an n × n matrix M n , its permanent is Per ( M n ) := P σ ∈ S n Q ni =1 m i,σ ( i ) . If M is a { , } matrix, then it is easy to see that the permanent counts the number of perfect matchings inthe bipartite graph with both parts of size n , where the i th vertex in the first part is connected to the j th vertex in the second part if and only if m i,j = 1 . It is well known (the upper bound is due to Bregman [3],solving the conjecture of Minc, and the lower bound was first obtained by Egorychev [7] and independently byFalikman [8], solving the conjecture of Van der Waerden) that if M is an n × n matrix with all entries either or whose row sums and column sums are all d , then Per ( M n ) = (cid:0) (1 + o (1)) de (cid:1) n . Therefore, starting with abipartite, d -regular graph H with parts of size n , by repeatedly removing perfect matchings from H , applying ∗ Massachusetts Institute of Technology. Department of Mathematics. Email: [email protected] . Research is partially sup-ported by an NSF grant 6935855. † Massachusetts Institute of Technology. Department of Mathematics. Email: [email protected] ‡ Department of Mathematics, ETH, 8092 Zurich, Switzerland. Email: [email protected].
Research supportedin part by SNSF grant 200021-175573. d ! ≈ (cid:0) (1 + o (1)) de (cid:1) d ), weget that there are (cid:0) (1 + o (1)) de (cid:1) nd -factorizations of H .For non-bipartite (regular) graphs, even deciding whether a single 1-factorization exists is usually a verydifficult question. For example, it took about years and an impressive tour de force of Csaba, Kühn,Lo, Osthus and Treglown [5] (improving an earlier asymptotic result of Perkovic and Reed [22]) to solve thefollowing old problem of Dirac: Theorem 1.1 ([5]) . Every d -regular graph G on n vertices, where n is a sufficiently large even integer and d ≥ ⌈ n ⌉ − , contains a -factorization. The above theorem is clearly tight in terms of d as can be seen, for example if n = 4 k + 2 , by taking G tobe the disjoint union of two cliques of size k + 1 = 2 ⌈ n ⌉ − (which is odd).Once an existence result is obtained, one can naturally ask for the number of distinct such structures.Given a d -regular graph G , it was shown by Kahn-Lovász (unpublished) and Alon-Friedland [1] that it hasat most ((1 + o (1)) d/e ) n/ perfect matchings. Therefore, the same reasoning as above (see also [19]) showsthat the number of -factorizations of G is at most (cid:0) (1 + o (1)) de (cid:1) dn/ . On the other hand, no matching lowerbounds were known for this problem. For the complete graph K n , Cameron [4] proved in 1976 that the numberof -factorizations is at least (cid:0) (1 + o (1)) n e (cid:1) n / (off by a factor of roughly − n / from the upper bound),which was recently improved by Ferber and Jain [9] to (cid:0) (1 + o (1)) n e (cid:1) n / . For general d -regular graphs with d ≥ n/ εn only weaker non-trivial lower bounds of the form n (1 − o (1)) dn/ are proven implicitly in [13] andin [11].In this paper, we give an asymptotically optimal lower bound for every d -regular graph G on n verticeswith d ≥ n + εn . That is, we prove the following: Theorem 1.2.
There exists a universal constant
C > such that for all sufficiently large even integers n andall d ≥ (1 / n − /C ) n , every d -regular graph G on n vertices has at least (cid:18)(cid:16) − n − /C (cid:17) de (cid:19) dn/ distinct 1-factorizations. Remark 1.3.
We have stated the above theorem in a stronger form ( − n − α instead of − o (1) ) with thehope that it might be useful in studying the behavior of typical -factorizations. That such a bound mightbe helpful was recently shown by Kwan [18] in the study of typical Steiner triple systems (which we do notdefine here).We conclude this introduction with a brief outline of the proof of our main result. Proof outline : Our proof is based on and extends ideas developed in [11] and [9], and largely goes as follows:First, we find an r -regular subgraph H ⊆ G , where r = d − τ , such that any ∆ -regular graph R ⊃ H with ∆ = (1 + o (1)) r contains a -factorization (we chose the letter R to denote the remainder graph obtainedafter deleting an ‘approximate’ -factorization from G ) . This is actually the key part of our argument andthe existence of such a graph (which, perhaps surprisingly, is quite simple!) is proven in Section 2.2.Next, we show that the graph G ′ which is obtained by deleting all the edges of H from G contains the‘correct’ number of ‘almost’ -factorizations. By an ‘almost’ -factorization, we mean a collection of edge-disjoint perfect matchings that cover almost all the edges of G ′ . This part is the most technical part of thepaper and is based on a suitable partitioning of the edge-set of G ′ into sparse subgraphs (quite similar to theone in [11]) along with a ‘nibbling’ argument from [6].Finally, for any given ‘almost’ -factorization of G ′ , by adding all the edges uncovered by this ‘almost’ -factorization to H , we obtain a graph R ⊃ H which is ∆ -regular with ∆ ≈ r , and therefore admits a -factorization by the property of H discussed above. Since the complete proofs are anyway not too long, wepostpone the more formal details to later sections. 2 Auxiliary results
In this section, we have collected a number of tools and auxiliary results to be used in proving our maintheorem.
Throughout the paper, we will make extensive use of the following well-known bounds on the upper andlower tails of the Binomial distribution due to Chernoff (see, e.g., Appendix A in [2]).
Lemma 2.1 (Chernoff’s inequality) . Let X ∼ Bin ( n, p ) and let E ( X ) = µ . Then • Pr [ X < (1 − a ) µ ] < e − a µ/ for every a > ; • Pr [ X > (1 + a ) µ ] < e − a µ/ for every < a < / . Sometimes, we will find it more convenient to use the following concentration inequality due to Hoeffding([15]).
Lemma 2.2 (Hoeffding’s inequality) . Let X , . . . , X n be independent random variables such that a i ≤ X i ≤ b i with probability one. If S n = P ni =1 X i , then for all t > ,Pr ( S n − E [ S n ] ≥ t ) ≤ exp (cid:18) − t P ni =1 ( b i − a i ) (cid:19) and Pr ( S n − E [ S n ] ≤ − t ) ≤ exp (cid:18) − t P ni =1 ( b i − a i ) (cid:19) . In this section we present the completion step, which uses some ideas from [9], and is a key ingredient ofour proof. Before stating the relevant lemma, we need the following definition.
Definition 2.3.
A graph H = ( A ∪ B, E ) is called ( α, r, m ) -good if it satisfies the following properties: ( G H is an r -regular, balanced bipartite graph with | A | = | B | = m . ( G Every balanced bipartite subgraph H ′ = ( A ′ ∪ B ′ , E ′ ) of H with | A ′ | = | B ′ | ≥ (1 − α ) m and with δ ( H ′ ) ≥ (1 − α ) r contains a perfect matching.The motivation for this definition comes from the next proposition, which shows that a regular graph onan even number of vertices, which can be decomposed into a good graph and a graph of ‘small’ maximumdegree, has a 1-factorization. Proposition 2.4.
There exists a sufficiently large integer m for which the following holds. Let m ≥ m , andsuppose that H = ( A ∪ B, E ( H )) is an ( α, r , m ) -good graph with m / ≤ r ≤ m and log m/r ≪ α < / .Then, for every r ≤ α r / log m , every r := r + r -regular (not necessarily bipartite) graph R on the vertexset A ∪ B , for which H ⊆ R , admits a 1-factorization.Proof. First, observe that e ( R [ A ]) = e ( R [ B ]) . Indeed, as R is r -regular, we have for X ∈ { A, B } that rm = X v ∈ X d R ( v ) = 2 e ( R [ X ]) + e ( R [ A, B ]) , from which the above equality follows. Moreover, ∆( R [ X ]) ≤ r for all X ∈ { A, B } since the only edges in R [ X ] come from R \ H . Next, let R := R and f := e ( R [ A ]) = e ( R [ B ]) . By Vizing’s theorem ([26]), both3 [ A ] and R [ B ] contain matchings of size exactly ⌈ f / ( r + 1) ⌉ . Consider any two such matchings M A in A and M B in B , and for X ∈ { A, B } , let M ′ X ⊆ M X denote a matching of size | M ′ X | = ⌊ αf / r ⌋ such that novertex v ∈ V ( H ) is adjacent to more than αr / vertices which are paired in the union of the two matchings.To show that such M ′ X must exist, it suffices to show that for X ∈ { A, B } , there exist matchings M ′′ X ⊆ M X with | M ′′ X | ≥ ⌊ αf / r ⌋ such that no vertex v ∈ V ( H ) is incident in H to more than αr / vertices whichare paired in the union of the two matchings, since we can obtain M ′ X from M ′′ X by simply removing theappropriate number of edges arbitrarily. Note that if the size of M X is at most αr / , then this followstrivially by taking M ′′ X = M X . When the size of M X is at least αr / , the existence of M ′′ X is seen usingthe following simple probabilistic argument. Let M ′′ X denote the random matching obtained by including eachedge of M X independently with probability α/ . By Chernoff’s inequality, | M ′′ X | ≥ α | M X | / > ⌊ αf / r ⌋ ,except with probability at most exp( − Θ( α | M X | )) ≤ exp( − Θ( α r )) ≪ , where the last inequality uses theassumption that α r ≫ log m ≫ . Moreover, each v ∈ V ( H ) is incident to at most r vertices, each of whichhas at most edge in M A ∪ M B . Hence, the number of vertices paired in M ′′ A ∪ M ′′ B incident to a fixed v ∈ V ( H ) is stochastically dominated by Bin ( r, α/ . In particular, by Chernoff’s inequality, the probability that afixed v ∈ V ( H ) is incident to at least αr / vertices matched in M ′′ A ∪ M ′′ B is at most exp( − Θ( αr )) ≪ /m ,where the last inequality uses the assumption that αr ≥ α r ≫ log m . Therefore, the union bound showsthat the above claim holds simultaneously for all vertices v ∈ V ( H ) with probability close to . It follows thatthere exist M ′′ A and M ′′ B with the desired properties.Delete the vertices in ( ∪ M ′ A ) S ( ∪ M ′ B ) , as well as any edges incident to them, from H and denote theresulting graph by H ′ = ( A ′ ∪ B ′ , E ′ ) . Since | A ′ | = | B ′ | ≥ (1 − α ) | A | and δ ( H ′ ) ≥ (1 − α/ r by the choice of M ′ X , it follows from ( G that H ′ contains a perfect matching M ′ . Note that M := M ′ ∪ M ′ A ∪ M ′ B is a perfectmatching in R . We repeat this process with R := R − M (deleting only the edges in M , and not the vertices)and f := e ( R [ A ]) = e ( R [ B ]) until we reach R k and f k such that f k ≤ r . Since f i +1 ≤ (1 − α/ r ) f i , thismust happen after at most r log m/α < α r steps. Moreover, since deg( R i +1 ) = deg( R i ) − , it follows thatduring the first ⌈ r log m/α ⌉ steps of this process, the degree of any R j is at least r − α r . Therefore, since ( r − α r ) − αr / ≥ (1 − α ) r , we can indeed use ( G throughout the process, as done above.From this point onwards, we continue the above process (starting with R k ) with matchings of size onei.e. single edges from each part, until no more edges are left. By the choice of f k , we need at most r suchiterations, which is certainly possible since r + 3 r log m/α < α r and ( r − α r ) − αr / ≥ (1 − α ) r .After removing all the perfect matchings obtained via this procedure, we are left with a regular, balanced, bipartite graph, which admits a 1-factorization (this follows by a more or less direct application of Hall’smarriage theorem [14]). Taking any such 1-factorization along with all the perfect matchings that we removedgives a 1-factorization of R .The remainder of this subsection is devoted to proving the following proposition, which shows that every d -regular graph on n vertices contains a ‘good’ subgraph H , assuming that d ≥ n/ εn and n is a sufficientlylarge even integer. Proposition 2.5.
Let n be a sufficiently large even integer, and let < ε = ε ( n ) < be such that ε ≫ log n/n .Let G be a graph on n vertices which is d -regular, with d ≥ n/ εn . Then, for every p = ω (cid:16) log nnε (cid:17) ≤ ,there exists a spanning subgraph H of G which is ( ε/ , r, n/ -good, for some (1 − ε/ dp/ ≥ r ≥ (1 − ε/ dp/ . To prove this proposition, we will use the following three results. The first result is a theorem from [10],which states that if G is a bipartite graph with sufficiently large minimum degree which contains an r -factor(i.e. a spanning r -regular subgraph) for r sufficiently large, then the random graph G p , which is obtained bykeeping each edge of G independently with probability p , typically contains a (1 − o (1)) rp -factor. The proofof this theorem follows quite easily by using the Gale-Ryser criterion for the existence of r -factors in bipartitegraphs ([12], [24]), and standard applications of Chernoff’s bounds. Theorem 2.6 (Theorem 1.4 in [10]) . Let m be a sufficiently large integer. Then, for any positive τ such that log m/m ≪ τ < , α = 1 / τ and < ρ ≤ α , the following holds. Suppose that: . G is bipartite with parts A and B , both of size m ,2. δ ( G ) ≥ αm , and3. G contains a ρm -factor.Then, for p = ω (cid:16) log mmτ (cid:17) , the random graph G p has a k -factor for k = (1 − τ ) ρmp with probability − m − ω (1) . Remark 2.7.
In [10], τ is taken to be some positive constant, as opposed to a function of m which can go to as m goes to infinity. However, the exact same proof actually gives the slightly more general result statedabove.The second result shows that if G is a bipartite graph with parts of size m , then with high probability, thenumber of edges in G p between subsets X and Y with | X | = | Y | ≤ m/ is not much more than pm | X | / . Lemma 2.8.
Let G = ( A ∪ B, E ) be a bipartite graph with parts A and B , both of size m . Let c = 1 / τ with < τ = τ ( m ) < , and let p = ω (cid:16) log mmτ (cid:17) . Then, for G p , the following holds with probability at least − m − ω (1) : e G p ( X, Y ) < cpm | X | for any subsets X ⊆ A and Y ⊆ B with | X | = | Y | ≤ m/ .Proof. Consider any subsets X ⊆ A and Y ⊆ B with | X | = | Y | ≤ m/ . Since e G ( X, Y ) ≤ | X || Y | ≤ | X | m , we get that Pr (cid:2) e G p ( X, Y ) ≥ cpm | X | (cid:3) ≤ Pr (cid:20) Bin (cid:18) m | X | , p (cid:19) ≥ cpm | X | (cid:21) ≤ exp( − τ m | X | p/ , where the first inequality follows from the fact that e G p ( X, Y ) is a sum of at most m | X | / independentBernoulli ( p ) random variables, and the second inequality follows from Chernoff’s bounds.Let B denote the event that there exist subsets X ⊆ A and Y ⊆ B with | X | = | Y | ≤ m/ and e G p ( X, Y ) ≥ cpm | X | . Then, it follows by the union bound thatPr [ B ] ≤ m/ X x =1 (cid:18) mx (cid:19) exp( − τ mxp/ ≤ m/ X x =1 (cid:16) emx (cid:17) x exp( − τ mxp/ ≤ m/ X x =1 exp(4 x log m − τ mxp/ ≤ m/ X x =1 exp ( x log m (4 − ω (1))) = m − ω (1) , where the inequality on the second line holds since pm = ω (log m/τ ) .Finally, the third result, which is a lemma from [11], shows that an almost regular bipartite graph withsufficiently large degrees contains an r -factor with r close to its minimum degree. Lemma 2.9 (Lemma 20 in [11]) . Let ρ ≥ / , m ∈ N , and ξ = ξ ( m ) > . Suppose that G = ( A ∪ B, E ) is abipartite graph with parts A and B , both of size m , and ρm + ξ ≤ δ ( G ) ≤ ∆( G ) ≤ ρm + ξ + ξ /m . Then, G contains a ρm -factor. We are now ready to prove Proposition 2.5.
Proof of Proposition 2.5.
Consider a random partitioning of V ( G ) with parts A, B of size m := | A | = | B | = n/ , and let G ′ = ( A ∪ B, E ′ ) denote the induced bipartite subgraph between A and B . For any v ∈ V ( G ) ,by linearity of expectation E [deg G ′ ( v )] = d d n − . A ⊆ V ( G ) coincides with the distribution on subsets of V ( G ) ob-tained by including every vertex independently with probability / , conditioned on the event (of probability Θ(1 / √ n ) ) that exactly n/ elements are included, it follows from Hoeffding’s inequality that for any fixed ver-tex v ∈ V ( G ) the probability that | deg G ′ ( v ) − E [deg G ′ ( v )] | ≥ √ d log n is at most O ( √ n exp( − d log n/d )) ≪ /n . Therefore, taking the union bound over all vertices, it follows that with high probability, d/ − p d log n ≤ δ ( G ′ ) ≤ ∆( G ′ ) ≤ d/ p d log n. Fix any such G ′ .We now wish to apply Lemma 2.9 to G ′ . For the choice of parameters, note that since d ≥ n/ εn , wecan find ρ ≥ / such that ρm = d/ − εm/ . If we take ξ = εm/ − √ d log n , then since ε ≫ log n/n ,we have ξ ≥ εm/ for all n sufficiently large, and hence ξ /m ≥ O ( ε m ) ≫ √ n log n . This shows that ρm + ξ ≤ δ ( G ′ ) ≤ ∆( G ′ ) ≤ ρm + ξ + ξ /m, Therefore, Lemma 2.9 applied to G ′ shows that G ′ contains a ρm = d/ − εm/ -factor. Note also that δ ( G ′ ) ≥ αm , with α = ρ + ( ξ/m ) ≥ / τ , where τ = ε/ .We will now apply Theorem 2.6 and Lemma 2.8 to G ′ in order to extract a sparse ‘good’ subgraph. Let p = ω (log m/ ( mε )) and consider the random graph G ′ p . Since G ′ satisfies the hypothesis of Theorem 2.6 withthe parameters ρ , α and τ as above, it follows that with high probability, G ′ p contains an r = (1 − τ ) ρmp =(1 − τ )(1 − εm/ d ) dp/ -factor H . In particular, note that (1 − ε/ dp/ ≤ (1 − τ )(1 − ε/ dp/ ≤ r ≤ (1 − τ ) dp/ − ε/ dp/ as desired. Moreover, applying Lemma 2.8 to G ′ with p and τ as above, wesee that with high probability, for any subsets X ⊆ A and Y ⊆ B , e G ′ p ( X, Y ) < (1 / ε/ pm | X | . Fixany such realization of G ′ p . We will show that H is ( ε/ , r, n/ -good.Suppose this is not the case. Then, by definition, there must exist a balanced bipartite subgraph H ′ =( A ′ ∪ B ′ , E ′ ) ⊆ H ⊆ G ′ p , with parts A ′ , B ′ of size | A ′ | = | B ′ | ≥ (1 − ε/ n/ and with δ ( H ′ ) ≥ (1 − ε/ r ,which does not contain a perfect matching. Therefore, by Hall’s marriage theorem (see, e.g., [25] for theversion used here), there must exist subsets X ′ ⊆ A ′ and Y ′ ⊆ B ′ with | X ′ | = | Y ′ | ≤ n/ such that at leastone of the following is true: N H ′ ( X ′ ) ⊆ Y ′ , or N H ′ ( Y ′ ) ⊆ X ′ . In either case, we get from the minimum degree assumption on H ′ that (1 − ε/ r | X ′ | ≤ e H ′ ( X ′ , Y ′ ) ≤ e G ′ p ( X ′ , Y ′ ) . Thus, we get e G ′ p ( X ′ , Y ′ ) ≥ (cid:16) − ε (cid:17) r | X ′ | ≥ (cid:16) − ε (cid:17) (cid:16) − ε (cid:17) dp | X ′ | ≥ (cid:16) − ε (cid:17) (cid:18)
12 + ε (cid:19) np | X ′ |≥ (cid:18)
12 + 3 ε (cid:19) np | X ′ | . However, since G ′ p satisfies the conclusion of Lemma 2.8, we must also have e G ′ p ( X ′ , Y ′ ) ≤ (1 / ε/ pm | X ′ | =(1 / ε/ np | X ′ | / , which leads to a contradiction. The following technical lemma allows us to partition our graph G into a number of smaller subgraphs, eachof which contains many ‘almost’ -factorizations. Its proof is similar to Lemma 27 in [11], but we need here adifferent set of parameters. Lemma 2.10.
Let n be a sufficiently large integer and let K be an integer in [log n, n / ] . Let τ > besuch that τ > /K . Suppose that G is a d -regular graph on n vertices with d ≥ n/ . Then, there are K edge-disjoint spanning subgraphs H , . . . , H K of G with the following properties: . For each H i , there is a partition V ( G ) = U i ∪ W i with | W i | = n/K ± ( n/K ) / and even;2. Letting F i = H i [ W i ] , we have δ ( F i ) ≥ ( d/n − τ ) | W i | ;3. Letting E i = H i [ U i , W i ] , we have e E i ( u, W i ) ≥ | W i | / K for all u ∈ U i ;4. Letting D i = H i [ U i ] for all i , then for some d/K ≥ r ≥ (1 − τ ) d/K , we have r ≤ δ ( D i ) ≤ ∆( D i ) ≤ r + r / . Proof.
First, let { S ( v ) } v ∈ V ( G ) be i.i.d random variables, where for each v ∈ V ( G ) , S ( v ) ⊆ [ K ] is a subset ofsize exactly K , chosen uniformly at random from among all such subsets. For each i ∈ [ K ] , let W i := { v ∈ V ( G ) : i ∈ S ( v ) } ; note that the graphs G [ W i ] are not necessarily edge-disjoint.Second, let s := n/K . Since any i ∈ [ K ] is included in S ( v ) with probability /K , independently fordifferent v ∈ V ( G ) , it follows by Chernoff’s bounds that the following holds for all v ∈ V ( G ) and i ∈ [ K ] with probability − o (1) : ( a ) | W i | = s (1 ± s − / ) ; ( b ) e G ( v, W i ) = dK (1 ± s − / ) .Next, for each v ∈ V ( G ) , we define the random variable Y ( v ) to be the number of vertices u ∈ N G ( v ) with { u, v } ⊂ W i for some i ∈ [ K ] . For each v ∈ V ( G ) and i ∈ [ K ] , we define the random variable Z i ( v ) to bethe number of vertices u ∈ N G ( v ) such that u ∈ W i and { u, v } ⊂ W j for some j ∈ [ K ] . Since all vertices of G have the same degree, the values of E [ Y ( v )] and E [ Z i ( v )] are the same for all choices of v and i . Let usdenote these common values by Y and Z , respectively.We claim that Y ≤ d/K and d/ K ≤ Z ≤ d/K for all sufficiently large K . To see this, note that Y = E [ Y ( v )] = E [ E [ Y ( v ) | S ( v )]] , and conditioned on any realization of S ( v ) , E [ Y ( v ) | S ( v )] ≤ X u ∈ N G ( v ) E S ( u ) [ | S ( u ) ∩ S ( v ) | ] = d E S ( u ) [ | S ( u ) ∩ S ( v ) | ] = d/K. Similarly, Z = E [ Z ( v )] = E [ E [ Z ( v ) | S ( v )]] . Conditioned on any realization of S ( v ) for which / ∈ S ( v ) , for allsufficiently large K , E [ Z ( v ) | S ( v )] = X u ∈ N G ( v ) Pr S ( u ) [ {| S ( u ) ∩ S ( v ) | > } ∩ { ∈ S ( u ) } ] ∈ [0 . d/K , . d/K ] , whereas conditioned on any realization of S ( v ) for which ∈ S ( v ) , E [ Z ( v ) | S ( v )] = d Pr S ( u ) [ {| S ( u ) ∩ S ( v ) | > } ∩ { ∈ S ( u ) } ] = d/K . Since Pr [1 ∈ S ( v )] = 1 /K , the desired conclusion follows from the law of total probability. Also, by Hoeffding’sinequality, it follows that with probability − o (1) , for all v ∈ V ( G ) and i ∈ [ K ] , ( c ) Y ( v ) = Y ± √ n log n ; ( d ) Z i ( v ) = Z ± √ n log n .Therefore, there exists a collection W , . . . , W K satisfying ( a ) , ( b ) , ( c ) and ( d ) simultaneously. Moreover, afterremoving at most one vertex from each W i , we may further assume that | W i | is even for all i ∈ [ K ] . Fix anysuch collection. Let U i := V ( G ) \ W i and let G ′ = ( V ( G ) , E ( G ′ )) , where E ( G ′ ) := E ( G ) \ ∪ i ∈ [ K ] E [ W i ] .To each edge e ∈ ∪ i ∈ [ K ] E [ W i ] , assign an arbitrary k ( e ) ∈ [ K ] such that e ⊂ W k ( e ) . Further, assignindependently to each edge e ∈ E ( G ′ ) , a uniformly chosen element k ( e ) ∈ [ K ] . For each i ∈ [ K ] , let7 i = ( V ( G ) , E ( H i )) , where E ( H i ) := { e ∈ E ( G ) : k ( e ) = i } . We claim that with probability at least − n − , H , . . . , H K satisfy the conclusions of the lemma.Conclusion . follows immediately from property ( a ) . For conclusion . , note that the only edges { u, v } ⊂ W i which are present in G [ W i ] but possibly not in H i [ W i ] are those which are also present in G [ W j ] for someother j ∈ [ K ] . Since for given i ∈ [ K ] and v ∈ V ( G ) , Z i ( v ) bounds the number of such edges incident to v ,it follows from properties ( b ) and ( d ) that for any i ∈ [ K ] and any v ∈ W i : deg F i ( v ) ≥ e G ( v, W i ) − Z i ( v ) ≥ e G ( v, W i ) − dK ≥ dK − dK . Therefore, property ( a ) shows that deg F i ( v ) ≥ dn | W i | − dK − dn n / K / ≥ (cid:18) dn − K (cid:19) | W i | . We now verify that conclusions . and . are satisfied with the desired probability. Property ( c ) shows thatfor all v ∈ V ( G ) , deg G ′ ( v ) = deg G ( v ) − Y ( v ) = d − Y ± p n log n. Moreover, properties ( b ) and ( d ) show that for all i ∈ [ K ] and for all u ∈ U i , e G ′ ( u, W i ) = e G ( u, W i ) − Z i ( u ) = dK (1 ± s − / ) − Z ± p n log n. Therefore, since each edge e in G ′ chooses a label k ( e ) ∈ [ K ] independently and uniformly, it follows byChernoff’s inequality that for all v ∈ V ( G ) , i ∈ [ K ] and u ∈ U i , deg H i ( v ) = deg G ′ ( v ) K ± p n log n = d − YK ± p n log ne H i ( u, W i ) = e G ′ ( u, W i ) K ± p n log n = dK (cid:18) − ZK d (cid:19) ± n / , except with probability at most (say) n − . Whenever this holds, we also get that for all i ∈ [ K ] and u ∈ U i , deg D i ( u ) = deg H i ( u ) − e H i ( u, W i ) = d − YK − dK (cid:18) − ZK d (cid:19) ± n / . This implies that for all i ∈ [ K ] and u ∈ U i , e E i ( u, W i ) ≥ dK − ZK − n / ≥ n K − dK − n / ≥ n K − nK − n / ≥ | W i | K − | W i | K − n / ≥ | W i | K , where the last line uses | W i | /K ≫ | W i | /K ≫ n / . We also have for all i ∈ [ K ] that δ ( D i ) ≥ dK − YK − dK + ZK − n / ≥ dK − dK − dK + d K − n / ≥ (cid:18) − K (cid:19) dK , and ∆( D i ) ≤ dK − YK − dK + ZK + 2 n / ≤ δ ( D i ) + 4 n / ≤ δ ( D i ) + δ ( D i ) / . Therefore, we may take (1 − τ ) d/K ≤ r ≤ d/K in conclusion 4. This completes the proof.8 emark 2.11. The proof of Lemma 2.10 given above actually shows that if we fix any collection W , . . . , W K satisfying properties ( a ) , ( b ) , ( c ) and ( d ) in the proof, then there are at least (cid:0) − n − (cid:1) ( K )( − K ) nd collections H , . . . , H K satisfying the conclusions of the lemma with respect to this choice of { U i , W i } K i =1 .This may be seen as follows: since we have seen that, given W , . . . , W K satisfying properties ( a ) , ( b ) , ( c ) and ( d ) , the random process to produce H , . . . , H K with the desired properties succeeds with probability at least − n − , it suffices to show that the number of outcomes of this random process is at least ( K )( − K ) nd .But this is immediate since | E ( G ′ ) | ≥ | E ( G ) | − K X i =1 | W i | ≥ dn − K · n K ≥ dn (cid:18) − K (cid:19) , and each edge e ∈ E ( G ′ ) chooses one of K labels. The fact that all these collections satisfy the conclusionof the lemma with respect to the same fixed choice of { U i , W i } K i =1 will be used crucially in the proof ofTheorem 1.2. The following proposition shows that an almost regular graph contains the ‘correct’ number of collectionsof large matchings such that every collection is equitable in the sense that each vertex is left uncovered byonly a small number of the matchings. The proof of this proposition follows from the proof of the main resultin the work of Dubhashi, Grable, and Panconesi [6]. For completeness, we include the details in Section 4,after the proof of our main result.
Proposition 2.12.
Let n be a sufficiently large even integer, and let G be a graph on n vertices and m edgeswith δ := δ ( G ) ≤ ∆( G ) =: ∆ such that δ ≥ n / and ∆ − δ ≤ ∆ / . There exists a universal constant J > for which the following holds. There are at least (cid:16) (1 − n − / J ) δe (cid:17) m distinct collections of edge-disjointmatchings M = { M , . . . , M δ } of G such that:1. Each matching M i covers at least (cid:0) − n − / J (cid:1) n vertices;2. Each vertex is uncovered by at most δ · n − / J matchings in M . In this subsection, we show how to complete a collection of edge-disjoint matchings of H [ U i ] into a collectionof edge-disjoint perfect matchings of G , using the sets W i . Lemma 2.13.
Let n be a sufficiently large even integer. Let K be an integer in [log n, n / ] and let H bea graph on n vertices for which:1. V ( H ) = U ∪ W with | W | even;2. | W | = nK ± ( nK ) / ;3. δ ( H [ W ]) ≥ (1 / τ / | W | with τ > /K ;4. Every vertex u ∈ U has at least nK edges into W .Let M be a collection of t ≤ nK edge-disjoint matchings of H [ U ] such that:(a) Every matching in M covers at least | U | − nK vertices of U ; b) Every vertex u ∈ U is uncovered by at most n/K matchings in M .Then, M can be extended to a collection of t edge-disjoint perfect matchings of H .Proof. Let M := { M , . . . , M t } be an enumeration of the matchings. For each M i , let C i ⊆ U denote the setof vertices which are not covered by M i . By assumption (a), we have | C i | ≤ n/K for all i ∈ [ t ] . We nowdescribe and analyze an iterative process to extend M , . . . , M t to edge-disjoint perfect matchings M , . . . , M t of H .Let H := H . For each u ∈ C , select a distinct vertex w ( u ) ∈ W such that { u, w ( u ) } is an edge in H .This is possible (and can be done greedily) since every u ∈ C has at least n/K > | C | edges into W byassumption 4. Let W ⊆ W denote the set of vertices in W which have not been matched to any vertex in C . Note that since | W | and | C | are even by assumption, | W | is also even. Moreover, | W | = | W | − | C | ≥ | W | − nK ≥ | W | (cid:18) − K (cid:19) , where the last inequality follows from assumption 2. and δ ( H [ W ]) ≥ δ ( H [ W ]) − ( | W | − | W | ) ≥ (cid:18)
12 + τ (cid:19) | W | − | W | K > | W | , where the second and third inequalities follow from assumption 3. A classical theorem due to Dirac showsthat any graph on k vertices with minimum degree at least k contains a Hamilton cycle, and hence, a perfectmatching; therefore, H [ W ] contains a perfect matching N . Let M := M ∪ N ∪ {{ u, w ( u ) } : u ∈ C } . Itis clear that M is a perfect matching in H . Continue this process starting with H , where H is the graphobtained from H by deleting the edges of M .To complete the proof, it suffices to show that the above procedure can be repeated t times. For this, wesimply need to observe two things. First, since each vertex u ∈ U is uncovered by at most n/K matchings M i by assumption (b), we need to use at most n/K edges from u into W during this process; in particular,at any stage i ∈ [ t ] during this process, every u ∈ U has at least n/K − n/K > | C i | edges into W . Second,since δ ( H i +1 [ W ]) ≥ δ ( H i [ W ]) − for all i ∈ [ t − and since t ≤ n/K ≤ τ | W | / , it follows that at anystage i ∈ [ t ] , δ ( H i [ W ]) ≥ (cid:18)
12 + τ (cid:19) | W | − t ≥ (cid:18)
12 + τ (cid:19) | W | , which is sufficient for the application of Dirac’s theorem as above. Remark 2.14.
The above proof shows that if M = { M , . . . , M t } and M ′ = { M ′ , . . . , M ′ t } are distinct collec-tions of t edge-disjoint matchings in H [ U ] satisfying the hypotheses of Lemma 2.13, then M = { M , . . . , M t } and M ′ = { M ′ , . . . , M ′ t } are distinct collections of t edge-disjoint perfect matchings in H . This is becausenone of the edges in M i \ M i and M ′ i \ M ′ i are present in H [ U ] , so that M i ∩ H [ U ] = M i and M ′ i ∩ H [ U ] = M ′ i . In this section we prove our main result, Theorem 1.2.
Proof of Theorem 1.2.
Let C = 2000 max { J, } where J is the constant appearing in the statement ofProposition 2.12. Our proof consists of two stages. In Stage 1, we describe our algorithm for construct-ing -factorizations. In Stage 2, we analyze this algorithm and show that it actually outputs the ‘correct’number of distinct -factorizations. Stage 1:
Our algorithm consists of the following five steps.10tep Let ε = n − /C and p = ε . Since ε = n − /C ≫ log n/n , p = ε = ω (log n/ε n ) ≤ , and d ≥ n/ εn ,it follows from Proposition 2.5 that there exists a spanning subgraph H of G which is ( ε/ , r , n/ -good, for some r = Θ( dp ) . Later (in Step 5.), we will apply Proposition 2.4 with H as the underlyinggood graph. For this, note that as required for this proposition, we indeed have that n / ≪ r =Θ( dp ) = Θ( nε ) ≪ n and log n ≪ r α = Θ( nε ) ≪ n .Step Let G ′ be the graph obtained from G by deleting all the edges of H . Then, G ′ is d ′ := ( d − r ) -regularand crucially, d ′ ≥ n/ εn/ , since r = Θ( dp ) = Θ( nε ) ≪ εn . For K = ⌊ ε − ⌋ , fix any collectionof K subsets of V ( G ′ ) = V ( G ) , denoted by W , . . . , W K , satisfying properties ( a ) , ( b ) , ( c ) and ( d ) asin the proof of Lemma 2.10 (which is applicable since K ∈ [log n, n / ] and d ′ ≥ n/ ). For i ∈ [ K ] ,let U i := V ( G ′ ) \ W i .Step Let H , . . . , H K be edge-disjoint spanning subgraphs of G ′ satisfying properties ., ., . and . in theconclusion of Lemma 2.10 for the choice of { U i , W i } K i =1 as above, and with τ = 200 /K . In particular,by conclusion . of Lemma 2.10, we have δ ( H i [ W i ]) ≥ (cid:18) d ′ n − τ (cid:19) | W i | ≥ (cid:18)
12 + 3 ε − τ (cid:19) | W i | ≥ (cid:18)
12 + ε (cid:19) | W i | (1)for all i ∈ [ K ] , where the last inequality holds since τ = Θ( K − ) = Θ( ε ) ≪ ε . Moreover, byconclusion . of Lemma 2.10, we also have that for all i ∈ [ K ] , d ′ K ≥ δ ( H i [ U i ]) ≥ (1 − τ ) d ′ K = (cid:18) − K (cid:19) d ′ K . Step Note that for each i ∈ [ K ] , we can apply Proposition 2.12 to the graph H i [ U i ] since δ i := δ ( H i [ U i ]) ≥ (1 − /K ) d ′ /K = Θ( ε n ) ≫ n / ≥ | U i | / and ∆( H i [ U i ]) − δ ( H i [ U i ]) ≤ (2∆( H i [ U i ])) / ≪ (∆( H i [ U i ])) / , where the first inequality follows from conclusion . of Lemma 2.10. Let M i denotea collection of matchings of H i [ U i ] satisfying the conclusions of Proposition 2.12. In particular, eachmatching M ∈ M i covers at least | U i | − | U i | − / J ≥ | U i | − nn / J ≥ | U i | − nK vertices, where the last inequality holds since K = Θ( ε − ) ≪ n / J , and each vertex u ∈ U i isuncovered by at most | U i | − / J δ i ≤ nK n / J ≤ nK matchings in M i , where the last inequality holds since K = Θ( ε − ) ≪ n / J .Note also that for each i ∈ [ K ] , we can apply Lemma 2.13 to H := H i and M := M i with τ = ε .Indeed, hypotheses . and . follow from conclusion . of Lemma 2.10, hypothesis . follows fromEquation (1), hypothesis . follows from conclusion . of Lemma 2.10 (since | W i | / K = Θ( n/K ) ≫ n/K ), hypotheses ( a ) and ( b ) follow from the computations earlier in this step, and finally, the numberof matchings in M i is at most δ i ≤ n/K . Hence, we can extend M i to a collection of edge-disjointperfect matchings of H i , which we will denote by M i .Step Let R be the graph consisting of all the edges in E ( G ′ ) which do not belong to any M i . Then, R isan r -regular graph with r = d ′ − K X i =1 δ i ≤ d ′ − d ′ (1 − /K ) = 200 d ′ /K. Since H is a ( ε/ , r , n/ -good graph (with α := ε/ and r satisfying the hypotheses of Proposition 2.4),and since r = 200 d ′ /K = Θ( nε ) ≪ α r / log n = Θ( nε / log n ) , we can apply Proposition 2.4 tothe graph H ∪ R in order to complete ∪ i ∈ [ K ] M i to obtain a -factorization of G .11 tage 2: We now show that the above algorithm can output the ‘correct’ number of distinct -factorizations.Throughout, we will assume that n is a sufficiently large even integer. By Remark 2.11, there are at least (cid:0) − n − (cid:1) ( K )( − K ) nd ′ = (cid:0) − n − (cid:1) (cid:16) K e −
30 log
K/K (cid:17) nd ′ / ≫ (cid:16) e −
30 log
K/K (cid:17) nd ′ / (cid:16) K e −
30 log
K/K (cid:17) nd ′ / ≥ (cid:16) K e −
60 log
K/K (cid:17) nd ′ / ≥ (cid:16) K e −
60 log
K/K (cid:17) nd (1 − p ) = (cid:16) K e −
60 log
K/K (cid:17) nd/ (cid:16) K − p e p log K/K (cid:17) nd/ ≫ (cid:16) K e − ε log K (cid:17) nd (cid:0) e − p log K (cid:1) nd ≥ (cid:0) K (cid:1) nd (cid:0) e − p log K (cid:1) nd distinct ways to choose the collection of subgraphs H , . . . , H K in Step 3. Moreover, by Proposition 2.12, foreach i ∈ [ K ] , there are at least (cid:18) (1 − | U i | − / J ) δ i e (cid:19) δi | Ui | ≥ (cid:18) (1 − n − / J ) δ i e (cid:19) δin (1 − /K ≥ (cid:18) (1 − n − / J )(1 − /K ) d ′ K e (cid:19) nd ′ (1 − /K )(1 − /K K ≥ (cid:18) (1 − ε ) d ′ K e (cid:19) nd ′ (1 − ε K ≥ (cid:18) (1 − ε )(1 − p ) dK e (cid:19) nd (1 − p )(1 − ε K ≥ (cid:18) (1 − p ) dK e (cid:19) nd K (1 − p ) ≥ (cid:18) (1 − p ) dK e (cid:19) nd K (cid:0) e − p log d (cid:1) nd K distinct ways to choose a collection of edge-disjoint matchings M i in H i [ U i ] satisfying the conclusions ofProposition 2.12. Since distinct collections of edge-disjoint matchings M i of H i [ U i ] stay distinct upon theapplication of Lemma 2.13 (see Remark 2.14), it follows that there are at least K Y i =1 (cid:18) (1 − p ) dK e (cid:19) nd K (cid:0) e − p log d (cid:1) nd K = (cid:18) (1 − p ) dK e (cid:19) nd (cid:0) e − p log d (cid:1) nd distinct ways to choose, one for each i ∈ [ K ] , a collection of edge-disjoint perfect matchings M i of H i as atthe conclusion of Step 4. of our algorithm in Stage 1. Together with the number of choices for H , . . . , H K in Step 3., it follows that the multiset of -factorizations of G that can be obtained by the algorithm in Stage1 has size at least (cid:18)(cid:18) (1 − p ) dK e (cid:19) K e − p log n (cid:19) nd ≥ (cid:18) (1 − p log n ) de (cid:19) nd . To complete the proof, it suffices to show that no -factorization F = { F , . . . , F d } is counted more than (1 + 400 p log n ) nd times in the calculation above. Let us call a collection of edge-disjoint subgraphs H , . . . , H K G ′ consistent with F if H , . . . , H K satisfy the conclusions of Lemma 2.10 with the parameters in Step 3(in particular, with respect to the fixed collection { U i , W i } K i =1 from Step 2), and if F can be obtained by thealgorithm after choosing H , . . . , H K in Step 3. It is clear that the number of times that F can be counted bythe above computation is at most the number of collections H , . . . , H K which are consistent with F , so thatit suffices to upper bound the latter. For this, note that at most r + 200 d ′ /K ≤ r of the perfect matchingsin F can come from Step 5 of the algorithm. Each of the other perfect matchings belongs completely to asingle H i by construction. It follows that the number of consistent collections H , . . . , H K is at most n · (cid:18) d r (cid:19) · (cid:0) K (cid:1) r n · (cid:0) K (cid:1) d . Indeed, there are at most n · (cid:0) d r (cid:1) ways to choose the perfect matchings coming from Step 5; these matchingscontain at most r n edges; for each such edge, there are at most K choices for which H i it should belong to;and for each of the remaining (at most d ) matchings which are completely contained in some H i , there are atmost K choices for which H i such a matching should belong to. Finally, observe that n · (cid:18) d r (cid:19) · (cid:0) K (cid:1) r n · ( K ) d ≪ d r K r n +3 d ≪ K r n ≪ K pnd = (cid:0) K p (cid:1) nd ≤ (1 + 400 p log n ) nd , which completes the proof. We now show how the proof of the main result in [6], which is based on the celebrated Rödl nibble[23], implies Proposition 2.12. The organization of this section is as follows: Algorithm 1 records the nib-bling algorithm used in [6]; Theorem 4.1 records the conclusion of the analysis in [6]; Proposition 4.3 adaptsTheorem 4.1 for our choice of parameters; Remark 4.4 shows that the collection of matchings produced byAlgorithm 1 satisfies the conclusions of Proposition 2.12, and finally, following this remark, we present theproof of Proposition 2.12.The following algorithm (Algorithm 1) is a slight variant of the algorithm used in [6] to find an almost-optimal edge coloring of a graph. Here, we show how it can be used to generate ‘almost the correct number’of ‘equitable collections of edge-disjoint large matchings’ of an ‘almost regular graph’ (all in the sense ofProposition 2.12). Since we are not concerned with the running time of the algorithm, we are able to make asimpler choice for the initial palettes of the edges as compared to [6]. Moreover, since our goal is to output alarge collection of edge-disjoint matchings as in Proposition 2.12, we have no need for the trivial ‘Phase 2’ ofthe algorithm in [6].The analysis of this algorithm is based on controlling the following three quantities: • | A i ( u ) | , the size of the implicit palette of vertex u at the end of stage i , where the implicit palette A i ( u ) denotes the set of colors not yet successfully used by any edge incident to u . • | A i ( e ) | , the size of the palette A i ( e ) of edge e at the end of stage i . Note that A i ( uv ) = A i ( u ) ∩ A i ( v ) . • deg i,γ ( u ) , the number of neighbors of u which, at the end of stage i , have color γ in their palettes.We record the outcome of their analysis as Theorem 4.1. Before stating it, we need some notation.Define d i and a i as follows. First, define initial values d , a := ∆ and then, recursively define d i := (1 − p τ ) d i − = (1 − p τ ) i ∆ lgorithm 1 The Nibble AlgorithmInput: The initial graph G := G on n vertices with minimum degree δ and maximum degree ∆ . Each edge e = uv is initially given the palette A ( e ) = { , . . . , δ } . For i = 0 , , . . . , t τ − stages, repeat the following: • (Select nibble) Each vertex u randomly and independently selects each uncolored edge incident to itselfwith probability τ / . An edge is considered selected if either or both of its endpoints selects it. • (Choose tentative color) Each selected edge e chooses independently at random a tentative color t ( e ) from its palette A i ( e ) of currently available colors. • (Check color conflicts) Color t ( e ) becomes the final color of e unless some edge incident to e has chosenthe same tentative color. • (Update graph and palettes) The graph and the palettes are updated by setting G i +1 = G i − { e | e got a final color } and, for each edge e , setting A i +1 ( e ) = A i ( e ) − { t ( f ) | f incident to e, t ( f ) is the final color of f } .a i := (1 − p τ ) a i − = (1 − p τ ) i ∆ = d i / ∆ , where p τ := τ (cid:16) − τ (cid:17) e − τ (1 − τ/ . In particular, note that setting t τ := 1 p τ log 4 τ , we have d t τ ≤ τ ∆ / . Theorem 4.1 ([6], Lemmas 10, 13 and 16, and the discussion in Section 5.5.) . There exist constants
K, c > such that, if at the end of stage i of Algorithm 1, the following holds for all vertices u , edges e and colors γ : | A i ( u ) | = (1 ± e i ) d i | A i ( e ) | = (1 ± e i ) a i deg i,γ ( u ) = (1 ± e i ) a i then, except with probability at most n − , the following holds at the end of stage i + 1 for all vertices u ,edges e and colors γ : | A i +1 ( u ) | = (1 ± e i +1 ) d i +1 | A i +1 ( e ) | = (1 ± e i +1 ) a i +1 deg i +1 ,γ ( u ) = (1 ± e i +1 ) a i +1 , where e i +1 = C ( e i + c p log n/a i ) = C ( e i + c (1 − p τ ) − i p log n/ ∆) , with C = 1 + Kτ . Remark 4.2.
In our case, we have | A ( u ) | = | A ( e ) | = δ = ∆ (cid:18) − ∆ − δ ∆ (cid:19) , deg ,γ = deg( u ) = ∆ (cid:18) ± ∆ − δ ∆ (cid:19) for all vertices u , edges e , and colors γ . Therefore, we can take e = ∆ − δ ∆ ≤ ∆ − / . Proposition 4.3.
Let n be a sufficiently large integer. There exists a constant J > for which the followingholds. Let G be a graph on n vertices with ∆ ≥ log n and e := (∆ − δ ) / ∆ ≤ ∆ − / , and let ∆ − /J ≤ τ < / . Then, with probability at least − t τ n − , the following holds for the execution of Algorithm 1 on G with parameter τ for t τ − stages: for all ≤ i ≤ t τ , and for all vertices u , all edges e , and all colors γ , • | A i ( u ) | = (1 ± τ ) d i • | A i ( e ) | = (1 ± τ ) a i • deg i,γ ( u ) = (1 ± τ ) a i Proof.
Setting A := c p log n/ ∆ and B := 1 / (1 − p τ ) , we see from Theorem 4.1 and the union bound that,except with probability at most t τ n − , e ℓ = C ℓ e + A [ C ℓ + C ℓ − B + · · · + CB ℓ − ] for all ≤ ℓ ≤ t τ . Since B = 1 / (1 − p τ ) ≤ K ′ τ for some constant K ′ > , it follows that e ℓ ≤ ℓ (1 + Lτ ) ℓ c p log n/ ∆ + (1 + Lτ ) ℓ e , where L = max { K, K ′ , c } . Since e ≤ ∆ − / and ∆ ≥ log n , it follows that e ℓ ≤ cℓ exp( Lτ ℓ )∆ − / . The right hand side is maximized when ℓ = t τ . Finally, since τ < / by assumption, we get that e t τ ≤ (cid:18) τ (cid:19) L ∆ − / ≤ τ , where the last inequality holds provided we take J ≥ L + 1) . Remark 4.4.
Consider any partial edge coloring of G satisfying the conclusions of Proposition 4.3, and let M := { M , . . . , M δ } denote the collection of edge-disjoint matchings of G obtained by letting M γ be theset of edges colored with γ . Then, | M γ | ≥ (1 − τ ) n/ for all γ ∈ [ δ ] . To see this, note that any vertex u which is not covered by M γ must have γ in its implicit palette A t τ ( u ) . Moreover, every vertex u has at least | A t τ ( u ) | > d t τ / missing colors, and therefore at least as many uncolored edges attached to it at the end ofstage t τ − . It follows that every vertex which is uncovered by M γ contributes at least d t τ / to the sum P u deg t τ ,γ ( u ) . Hence, if n γ denotes the number of vertices uncovered by M γ , then n γ d t τ ≤ X u deg t τ ,γ ( u ) . On the other hand, we have X u deg t τ ,γ ( u ) ≤ a t τ n ≤ d t τ n ∆ . Combining these two inequalities and using d t τ ≤ τ ∆ / , we see that n γ ≤ τ n , as desired.Moreover, as mentioned above, the number of times a given vertex u is left uncovered by a matching in M equals | A t τ ( u ) | , which is at most d t τ ≤ τ ∆ / . Below, in the proof of Proposition 2.12, we will take τ := ∆ − /J . Using this choice of parameter in the two estimates in the present remark, it follows readily thatthat M satisfies the conclusions of Proposition 2.12. 15e are now ready to prove Proposition 2.12. Proof of Proposition 2.12.
We will view the execution of Algorithm 1 as a branching process where at eachstage, we branch out according to which edges get assigned final colors, and which final colors are assigned tothese edges. In particular, the leaves of this branching process are at distance t τ − from the root.We say that a leaf L of this branching process is a good leaf if the unique path from the root to L represents an execution of Algorithm 1 such that at all stages, all vertices u , all edges e , and all colors γ satisfy the conclusions of Proposition 4.3. Also, we say that a partial coloring with δ colors is a good partialcoloring if the corresponding collection of edge-disjoint matchings M = { M , . . . , M δ } satisfies the conclusionsof Proposition 2.12. Note by Remark 4.4 that the partial edge coloring corresponding to a good leaf is a goodpartial coloring. Therefore, in order to lower bound the number of good partial colorings, it suffices to lowerbound the number of good leaves, and upper bound the number of distinct leaves any given good partialcoloring can correspond to.For this, let Q > be an upper bound for the probability of the branching process reaching a given goodleaf. Since, by Proposition 4.3, the probability that the branching process reaches some good leaf is at least − t τ n − , it follows that the number of good leaves is at least (1 − t τ n − ) /Q . Further, let R > be suchthat for any good partial coloring C , there are at most R leaves of the branching process whose correspondingpartial coloring is C . Then, it follows that the number of good partial colorings is at least (1 − t τ n − ) /QR .The remainder of the proof consists of upper bounding the quantity QR .To upper bound Q , fix a good leaf L and note that in the i th stage of the execution corresponding to L , m i specific edges must be selected and assigned their final colors, where m i : = (1 ± τ ) d i n − (1 ± τ ) d i +1 n d i n (cid:0) ± τ − (1 − p τ )(1 ± τ ) (cid:1) = d i n (cid:0) p τ ± τ (cid:1) = τ d i n ± τ ) , and the last line follows since τ < / . Since each edge is selected independently with probability τ (1 − τ / (an edge is selected if and only if it is selected by at least one of its endpoints, which happens with probability τ / τ / − τ / by the inclusion-exclusion principle), and each selected edge chooses one of at least (1 − τ ) a i colors uniformly at random, it follows that the probability that the branching process makes the specificchoices at the i th stage of L is at most (cid:18) τ (1 − τ )(1 − τ ) a i (cid:19) m i (cid:16) − τ (cid:16) − τ (cid:17)(cid:17) (1 − τ din − m i ≤ (cid:18) τ (1 + τ ) a i (cid:19) m i (cid:16) − τ (cid:16) − τ (cid:17)(cid:17) (1 − τ din − m i ≤ (cid:18) τ (1 + τ ) a i (cid:19) m i (cid:16) − τ (cid:16) − τ (cid:17)(cid:17) (1 − τ din − (1 ± τ ) τdin ≤ (cid:18) τ (1 + τ ) a i (cid:19) m i exp (cid:18) − τ (cid:16) − τ (cid:17) d i n (cid:19) (1 − τ ) − (1 ± τ ) τ ≤ (cid:18) τ (1 + τ ) a i (cid:19) m i exp (cid:18) − (1 − τ ) τ d i n (cid:19) , where the first and last inequalities use τ < / . Since the randomness in different stages of the branchingprocess is independent, it follows that Q ≤ t τ − Y i =0 (cid:18) τ (1 + τ ) a i (cid:19) m i exp (cid:18) − (1 − τ ) τ d i n (cid:19) = exp − (1 − τ ) τ n t τ − X i =0 d i ! t τ − Y i =0 (cid:18) τ (1 + τ ) a i (cid:19) m i = exp (cid:18) − (1 − τ )(1 ± τ ) ∆ n (cid:19) t τ − Y i =0 (cid:18) τ (1 + τ ) a i (cid:19) m i = exp (cid:18) − (1 ± τ ) ∆ n (cid:19) t τ − Y i =0 (cid:18) τ (1 + τ )∆(1 − p τ ) i (cid:19) m i = exp (cid:18) − (1 ± τ ) ∆ n (cid:19) (cid:18) τ (1 + τ )∆ (cid:19) P tτ − i =0 m i (1 − p τ ) − P tτ − i =0 im i exp (cid:18) − (1 ± τ ) ∆ n (cid:19) (cid:18) τ (1 + τ )∆ (cid:19) (1 ± τ ) τn P tτ − i =0 d i (1 − p τ ) − P tτ − i =0 im i = exp (cid:18) − (1 ± τ ) ∆ n (cid:19) (cid:18) τ (1 + τ )∆ (cid:19) (1 ± τ )∆ n (1 − p τ ) − P tτ − i =0 im i = exp ( − (1 ± τ ) m ) (cid:18) τ (1 + τ )∆ (cid:19) (1 ± τ ) m (1 − p τ ) − P tτ − i =0 im i , where we have used that t τ − X i =0 d i = ∆ t τ − X i =0 (1 − p τ ) i = (1 ± τ ) ∆ τ in the third and seventh lines, τ < / in the fourth and seventh lines, and m = (1 ± τ )∆ n/ in the lastline.To upper bound R , it suffices to upper bound the number of ways in which the edges of any good partialcoloring can be partitioned into sets of size { (1 ± τ ) τ d i n/ } t τ − i =0 , since the number of edges colored bythe algorithm in the i th stage is (1 ± τ ) τ d i n/ . For this, note that there are at most (10 τ ∆ n ) t τ waysto choose the sizes of these t τ sets, and for each such choice for the sizes of the sets, there are at most m ! / Q t τ − i =0 ((1 − τ ) m i )! ways to partition the edges into these sets. Therefore, R (10 τ ∆ n ) − t τ ≤ m ! Q t τ − i =0 ((1 − τ ) m i )! ≤ m (cid:16) me (cid:17) m t τ − Y i =0 (cid:18) e (1 − τ ) m i (cid:19) (1 − τ ) m i = m (cid:16) me (cid:17) m t τ − Y i =0 (cid:18) e (1 − τ )(1 ± τ ) τ d i n (cid:19) (1 − τ ) m i = m (cid:16) me (cid:17) m t τ − Y i =0 (cid:18) e (1 − τ ) τ ∆(1 − p τ ) i n (cid:19) (1 − τ ) m i = m (cid:16) me (cid:17) m (cid:18) e (1 − τ ) τ ∆ n (cid:19) (1 − τ ) P tτ − i =0 m i t τ − Y i =0 (1 − p τ ) − im i (1 − τ ) = m (cid:16) me (cid:17) m (cid:18) e (1 − τ ) τ ∆ n (cid:19) (1 ± τ )∆ n (1 − p τ ) − (1 − τ ) P i im i ≤ m (cid:16) me (cid:17) m (cid:18) e (1 − τ ) τ m (cid:19) (1 ± τ ) m (1 − p τ ) − (1 − τ ) P i im i ≤ (cid:16) me (cid:17) τm (cid:18) (1 ± τ ) τ (cid:19) m (1 ± τ ) (1 − p τ ) − (1 − τ ) P i im i ≤ ( me ) τm (cid:18) τ (cid:19) m (1 ± τ ) (1 − p τ ) − (1 − τ ) P i im i , where the second line uses the standard approximation ( k/e ) k ≤ k ! ≤ k ( k/e ) k for k ≥ ; the third lineuses the definition of m i ; the fourth line uses the definition of d i along with τ < / ; the sixth line uses P t τ − i =0 m i = (1 ± n/ as shown in the calculation of the upper bound on Q ; the seventh line uses m = (1 ± τ )∆ n/ ; the eighth line uses ∆ − /J < τ < / , and the last line uses (1 + 50 τ ) ≤ e τ and τ < / .It follows that QR ≤ (10 τ ∆ n ) t τ ( me ) τm exp ( − (1 − τ ) m ) (cid:18) τ (1 + τ )∆ (cid:19) (1 ± τ ) m (cid:18) τ (cid:19) m (1 ± τ ) (1 − p τ ) − P i im i ≤ (10 τ ∆ n ) t τ (cid:18) me ∆ τ (cid:19) τm (cid:18) e ∆ (cid:19) m (1 − p τ ) − P i im i (10 τ ∆ n ) t τ (cid:18) me ∆ τ (cid:19) τm (cid:18) eδ (cid:19) m e p τ (1+10 τ ) P tτ − i =0 im i ≤ m t τ ( eδ ) − m (cid:18) me ∆ τ (cid:19) τm e ± τ ) τ ∆ n pτ ≤ m τm ( eδ ) − m (cid:18) me ∆ τ (cid:19) τm e ± τ ) m ≤ ( eδ ) − m e m (cid:18) me ∆ τ (cid:19) τm = (cid:18) e δ (cid:19) m e τm log( me ∆ /τ ) ≤ (cid:18) e τ log( m/τ ) e δ (cid:19) m , where the second line uses the inequality (1 + τ ) ≤ e τ ; the third line uses the inequality (1 − p τ ) ≥ e − p τ (1+10 τ ) which holds since τ < / ; the fourth line uses the following computation: t τ − X i =0 im i = (1 ± τ ) τ n t τ − X i =0 id i = (1 ± τ ) τ ∆ n t τ − X i =0 i (1 − p τ ) i ≤ (1 ± τ ) τ ∆ n ∞ X i =0 i (1 − p τ ) i = (1 ± τ ) τ ∆ n p τ , along with τ < / , and the fifth line uses m = (1 ± τ )∆ n/ along with τ < (1 + 5 τ ) p τ and t τ < /τ < τ m which holds since m − /J ≤ τ < / . Finally, substituting τ = ∆ − /J completes the proof. • We proved that the number of -factorizations in a d -regular graph is at least (cid:18) (1 + o (1)) de (cid:19) dn/ , provided that d ≥ n + εn . As mentioned in the introduction, this is asymptotically best possible. It willbe very interesting to obtain a similar result for all d ≥ ⌈ n/ ⌉ − (the existence of a -factorization inthis regime was proven in [5]). • As mentioned in Remark 1.3, we obtain an explicit function (polynomial in n ) for the (1 + o (1)) -term inthe bound on the number of -factorizations. We have written such a formula with the hope that it couldbe useful towards studying the behavior of typical -factorizations. For example, using similar bounds onthe number of Steiner triple systems as obtained by Keevash [16], Kwan [18] was recently able to studysome non-trivial properties of typical Steiner Triple Systems. Therefore, we hope that building uponKwan’s ideas and using our counting argument one could obtain some non-trivial properties of typical -factorizations. For example, can one show that a typical -factorization of K n contains a rainbow Hamiltonian path? (that is, a Hamiltonian path which uses exactly one edge from each of the perfectmatchings). • Another very interesting direction is to study the number of -factorizaitons in hypergraphs. In thissetting, much less is known and every non-trivial lower bound on the number of such factorizations shouldrequire new ideas. We are curious whether one can attack this problem using some clever reduction tothe graph setting and use our ideas for the ‘completion part’.18 cknowledgment: The first author is grateful to Kyle Luh and Rajko Nenadov for helpful discussions atthe first step of this project. We would also like to thank the anonymous referees for a very thorough readingof the manuscript and numerous invaluable comments.
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