EDGE-DISJOINT CLIQUES IN GRAPHS WITH HIGH MINIMUM DEGREE

Transcription

1 EDGE-DISJOINT CLIQUES IN GRAPHS WITH HIGH MINIMUM DEGREE RAPHAEL YUSTER Abstrat For a graph G and a fixed integer k 3, let ν k G) denote the maximum number of pairwise edge-disjoint opies of K k in G For a onstant, let ηk, ) be the infimum over all onstants γ suh that any graph G of order n and minimum degree at least n has ν k G) γn o n )) By Turán s Theorem, ηk, ) = 0 if /k ) and by Wilson s Theorem, ηk, ) /k k) as We prove that for any > > /k ), ηk, ) k) ) k Π k i= i + ) i) + k) ) k while it is onjetured that ηk, ) = /k k) if k/k + ) and ηk, ) = / k /k if k/k + ) > > /k ) The ase k = 3 is of partiular interest In this ase the bound states that for any > > /, η3, ) 4 By further analyzing the ase k = 3 we obtain the improved lower bound 5 + ) ) η3, ) 3 ) This bound is always at most within a fration of 0 38)/6 > 076 of the onjetured value whih is η3, ) = /6 for 3/4 and η3, ) = / /4 if 3/4 > > / Our main tool is an analysis of the value of a natural frational relaxation of the problem Keywords: triangles, paking, frational Introdution All graphs onsidered here are finite and simple undireted graphs For standard graph-theoreti terminology the reader is referred to [] The problem of omputing a maximum set of pairwise edge-disjoint subgraphs of a graph G that are isomorphi to a given fixed graph H is a fundamental problem in extremal graph theory and in design theory Perhaps the most studied ase is when H is the omplete graph H = K k, dating bak to a lassial result of Kirkman [5] See [3, 4, 8, 9,, 4,, 8,, 3, 4] for some representative works in this area For a graph G and a fixed integer k 3, let ν k G) denote the maximum number of pairwise edge-disjoint opies of K k in G If G is suffiiently sparse, then we might have ν k G) = 0 In fat, Turán s Theorem states that ν k G) = 0 only if eg) tn, k ) Here tn, r) denotes the number of edges of the graph T n, r) whih is the omplete r-partite graph with n verties and either n/r or n/r verties in eah partite lass As the minimum degree of T n, r) is at most n /r), a graph G with minimum degree n might have ν k G) = 0 when /k ) On the other hand, the simple greedy algorithm shows that ν k G) = Θn ) for any onstant > /k ) In fat, by Wilson s Theorem [3], K n has a set of n / k edge-disjoint opies of Kk as long as n is suffiiently large and satisfies some neessary divisibility onditions Hene, when, graphs with minimum degree at least n have ν k G) lose to n ) / k ) It is therefore of interest to determine νk G) as a funtion of the minimum degree of G Department of Mathematis, University of Haifa, Haifa 3905, Israel E mail: raphy@mathhaifaail

2 More formally, for a real onstant and an integer k 3, let ηk, ) be the infimum over all onstants γ suh that any graph G with n verties and minimum degree at least n has ν k G) γn o n )) By the disussion in the previous paragraph, Turán s Theorem states that ηk, ) = 0 if /k ) and, more generally, ηk, ) 0 as /k ) On the other hand, Wilson s Theorem asserts that ηk, ) /k k) as Observe also the trivial upper bound ηk, ) /k k) sine a n-regular graph annot have more than n / pairwise edge-disjoint opies of K k A longstanding onjeture of Nash-Williams [8] states that η3, ) = /6 for all 3/4 In fat, the onjeture of Nash-Williams is sharper in the sense that if, in addition, some divisibility onditions hold the degree of eah vertex is even and the overall number of edges is a multiple of 3), then a graph with minimum degree at least 3n/4 has a triangle deomposition Gustavsson, in his PhD Thesis [7] proved that η3, ) = /6 for 0 4 A speial ase of a reent breakthrough paper of Keevash [3] gives an alternative proof to Gustavsson s result The results of Gustavsson and Keevash, just like the onjeture of Nash-Williams, are deomposition results The value of for whih η3, ) = /6 was improved by the author in [4] to /0 5 and reently by Dukes [] to /6 The onjeture of Nash Williams is sharp in the sense that it annot be improved A simple interpolation argument provides a onstrution showing that η3, ) / /4 for / 3/4 We state this upper bound as the onjetured value for η3, ) Conjeture 3 6 if 4 <, η3, ) = 4) if 3 4, 0 if 0 As shown in Gustavsson s thesis, it is not diffiult to generalize the onstrution of Nash Williams, and his onjeture, to larger values of k The analogous onjeture in this ase is that ηk, ) = /k k) for all /k+) An interpolation argument, given in Setion 7, provides a onstrution showing that ηk, ) / k /k for /k ) /k + ) We state this upper bound as the onjetured value for ηk, ) Conjeture k k if k+ <, ) ηk, ) = k k 0 if 0 if k k+, k Our main results are lower bounds for ηk, ) in general and ηk, 3) in partiular As far as we know, these are the first nontrivial lower bounds that apply to the entire spetrum of Our first main result is a general lower bound for ηk, ) Theorem 3 Let k 3 be an integer and let be a real satisfying /k ) < < Then ηk, ) ) k ) k Π k i= i + ) i) + k ) ) k Observe first that if > /k ), then Theorem 3 asserts that ηk, ) > 0 and if, then Theorem 3 asserts that ηk, ) /k k), whih is onsistent with

3 the disussion above The following orollary is a restatement of Theorem 3 for the ases k = 3 and k = 4, given for referene Corollary 4 ) η3, ) 4 for > >, η4, ) for > > 3 Let us ompare ) to some other lower bounds that an be derived from existing results A lassial result of Goodman [6] on the triangle densities of graphs, states that a graph with n edges has ) n 3) on 3 ) triangles From this fat, together with the fat that K n has nn )/6 on) edge-disjoint triangles, one easily obtains that a graph with n edges has )n /6 on ) edge-disjoint triangles Thus, the easy lower bound η3, ) /3 /6 follows immediately Observe that the latter bound only relies on the fat that the edge density of the graph is and does not use the assumption that the minimum degree is n If we do use the minimum degree ondition, together with an observation that one an assume that any edge is ontained in at most n triangles see more details on the latter assumption in Setion 3) and together with the fat that, in dense graphs, a maximum frational triangle paking has asymptotially the same value as a maximum integral one see details on this fat in Setion, then Goodman s result implies that η3, ) /3 /6 However, observe that the lower bound ) is always better than /3 /6 for > > / It is important to note that Goodman s result on the density of triangles is optimal for values of of the form /t where t is an integer, but is not optimal for other values of The optimal bound was reently obtained in a breakthrough paper of Razborov [0] If we use Razborov s bound instead of Goodman s bound for values of not of the form /t), then the bound /3 /6 is slightly improved, but still falls short of our lower bound ) for all > > / As an illustrative example, onsider the ase = 5/8 Goodman s bound would then imply η3, 5/8) /4 = 5/0 Razborov s bound would imply η3, 5/8) 5/08, while ) shows that η3, 5/8) 5/96 Finally, note that Conjeture states that η3, 5/8) = /6 = 6/96 A generalization of Goodman s result for arbitrary k 3 was obtained by Lovász and Simonovits [6] see also [7]) Nikiforov [9] generalized Razborov s result to k = 4 Their bound an be used to prove a lower bound for ηk, ) using a similar reasoning as for triangles Again, the obtained bound using this approah is always smaller than the one given in Theorem 3 For example, if k = 4, then there are always )3 n 4) on 4 ) opies of K 4 This an be used to give the bound η4, ) )3 / However, observe that the lower bound is always better than )3 / for > > /3 Theorem 3 is a general lower bound for all k and However, for the ase k = 3, it is possible to obtain a non-negligible improvement using a more detailed analysis This is our seond main result Theorem 5 Let be a onstant satisfying / < < Then ) ) 3) η3, ) 3 ) For /, /3) the density of triangles was established already in [5] 3

4 Observe that if > /, then Theorem 3 asserts that η3, ) > 0 and if, then Theorem 3 asserts that η3, ) /6, whih is onsistent with the disussion above As shown in Setion 6, the bound 3) is always better than the bound ) for > > / One should therefore ompare 3) with Conjeture reall that the onjetured bound is also an upper bound) As shown in Setion 6, the ratio between the lower bound 3) and the onjetured bound is never smaller than 0 38)/6 > 076 for any > > / To prove our main results, we first redue the problem of omputing ηk, ) to the problem of omputing its frational relaxation The fat that suh redutions arry asymptotially no loss is an important result of Haxell and Rödl [] We then onsider a partiular frational assignment that we all the natural frational paking We prove that the natural frational paking always attains the values laimed in Theorem 3 and Theorem 5 Proving this requires some detailed ombinatorial and analytial onsiderations The rest of this paper is organized as follows In Setion we review the redution from integral to frational pakings and define the notion of a natural frational paking Setion 3 ontains the proof of Theorem 3 for the ase of triangles The general ase of Theorem 3, whih ontains signifiantly more notation but otherwise uses the same ideas as given in Setion 3 is proved in Setion 4 The proof of the improved bound for η3, ) namely, Theorem 5) appears in Setion 5 The quality of the lower bounds obtained in Theorems 3 and 5 with respet to the onjetured values for k = 3, 4 is established in Setion 6 Construtions of the upper bounds that oinide with Conjeture are provided in Setion 7 The final setion ontains some onluding remarks Preliminaries Let G H) denote the set of opies of a graph H in a graph G A funtion ϕ from ) G H to [0, ] is a frational H-paking of G if H H) G : e H ϕh ) for eah e EG) For a frational H-paking ϕ, let ϕ = H H) G ϕh ) The frational H-paking number, denoted by νh G), is the maximum value of ϕ ranging over all frational H-pakings ϕ An H-paking of G is a set of pairwise edge-disjoint opies of H in G Let ν H G) denote the maximum size of an H-paking of G In the ase H = K k we set ν k G) = ν Kk G) and νk G) = ν K K G) Trivially, νh G) ν HG), as an H-paking is a speial ase of a frational H- paking An important result of Haxell and Rödl [] see also [5]), whih relies on Szemerédi s regularity lemma [], shows that the onverse is also asymptotially true, up to an additive error term whih is negligible for suffiiently dense graphs Lemma For every ϵ > 0 and every graph H there exists N = NH, ϵ) suh that for all n > N, if G is a graph with n verties, then νh G) ν HG) ϵn For a real onstant and an integer k 3, let η k, ) be the infimum over all onstants γ suh that any graph G with n verties and minimum degree at least n has νk G) γn o n )) The following is an immediate orollary of Lemma Corollary η k, ) = ηk, ) For an edge e EG) let f H e) denote the number of elements of G H) that ontain e We sometimes omit the subsript H if it an be determined from ontext Let ψ = ψ H G) be the frational H-paking of G defined by ψx) = max e EX) f H e) In other words, for eah X G H), we look at all the edges of X, take the edge that 4

5 appears in the largest amount of elements of G H), and assign to X the value whih is reiproal to this amount We all ψ the natural frational H-paking of G Observe that ψ is indeed a valid frational H-paking of G as for eah edge, the sum of the weights of the opies of H ontaining it is at most f H e) /f H e)) = It may be that ψ H G) < νh G) For example, let G = K 5 and let H = K 3 In this ase, G H) ontains 7 triangles Eah triangle ontains at least one edge that appears in three triangles, so eah triangle reeives the weight /3 in the natural frational triangle paking of K5 The value of ψ is therefore 7/3 On the other hand, it is easy to verify that ν3k 5 ) = 3 The proofs of Theorems 3 and 5 are based on lower-bounding the value of the natural frational K k -paking of a graph with minimum degree n, thereby establishing a lower bound for η k, ) By Corollary, this also establishes a lower bound for ηk, ) 3 Proof of Theorem 3 for triangles Let > > / be fixed Let G be a graph with n verties and minimum degree δg) n By Corollary, it suffies to prove that ψ, the natural frational triangle paking of G, satisfies ) ψ n 4 We may assume that for any edge u, v) EG), either du) = δg) or dv) = δg), as otherwise we may remove u, v) and prove the theorem for the resulting subgraph, whih still has the same minimum degree Let m denote the number of edges of G and observe that m n / Reall that for an edge e EG), fe) = f K3 e) denotes the number of triangles ontaining e Hene, by our assumption that any edge is inident with a vertex of degree δg) and by the fat that the number of ommon neighbors of any two verties is at least δg) n, we have 3) )n fe) δg) < n Let T G) denote the set of triangles of G and let α = T G) /n 3 For a triangle T T G), we have that ψt ) = Consider first the ase where max e ET ) fe) n α 3 4 Observe that indeed α > 0 sine > / In this ase we easily have 3 ψ = ) ψt ) αn 3 n n 4 T T G) Consider from here onwards the ase that 0 < α / 3 /4 ) Order EG) as a sequene e,, e m where fe i ) fe i+ ) for i =,, m Now onsider the non-dereasing sequene B = {b, b, } of 3αn 3 = 3 T G) rationals, onsisting of m bloks, where the first fe ) elements of B are /fe ) this is the first blok), 5

6 the next fe ) elements are /fe ) the seond blok), and so on, where the final blok onsists of fe m ) elements whih are /fe m ) The sum of the elements of B is, therefore, m, sine the sum of eah blok is Observe that any element of the sequene B an be mapped to a triangle T T G) Indeed, the fe j ) elements of B in blok j are bijetively mapped to the fe j ) triangles ontaining e j Denote this mapping by M : B T G) Also notie that for eah T T G) there are preisely three elements of B that are mapped to T, one for eah edge of T The first of these three elements in the sequene B is alled the leading element with respet to T Suppose that ET ) = {e i, e j, e k } and that fe i ) fe j ) fe k ) The leading element with respet to T has value /fe i ) = ψt ) Let B be the subsequene of B onsisting of the leading elements, one for eah T T G) By definition of ψ, and sine eah element of B orresponds to the weight given by ψ to a triangle, we have that the sum of the elements of B is preisely ψ But B is some subsequene of T G) elements of the nondereasing sequene B, so in partiular, its sum is at least the sum of the first T G) = αn 3 elements of B Hene, ψ b l αn 3 l= Summarizing, we have that B is a nondereasing sequene of 3αn 3 rationals, whose sum is m where m n / Eah element in the sequene is at least /n) and at most / )n), and we wish to lower bound the minimum possible sum of the first αn 3 elements of this sequene Let therefore S = {s, s, } be any nondereasing sequene of 3αn 3 rationals, whose sum is m, and whose elements are between /n) and / )n) and for whih the sum of the first αn 3 elements is minimized We may assume that all the first αn 3 elements of S are equal, as otherwise we an just replae eah of them with their average So, we may assume that the first αn 3 elements of S are all equal to some value /xn), and that the average value of the remaining αn 3 elements of S is some value /yn), and reall that we have /yn) / )n) It follows that m = 3αn 3 i= s i = αn 3 xn + αn3 yn Using m n / and y we obtain from the last equation that Hene, 33) s i = αn 3 αn 3 i= α x + α xn α ) n n / 3 ) /4 )) ) = n 4 6

7 By the minimality of S we have that as required ψ b i s i αn 3 i= αn 3 i= ) n 4 4 Proof of Theorem 3 The proof of Theorem 3 an be generalized to other fixed omplete graphs K k by lower-bounding the natural frational K k -paking We outline the differenes between the proof for K 3 given in the previous setion and its more general form given here The first modifiation is equation 3) whih gives upper and lower bounds for fe) Now fe) is the number of K k that ontain e Clearly, fe) is at most the number of opies of K k in the ommon neighborhood of u and v where e = u, v) An upper bound for fe) follows from the fat that at least one of the endpoints has degree δg) n So, there ould be at most n k opies of Kk in the ommon neighborhood of u and v Thus, fe) n) k /k! On the other hand, u and v has at least )n ommon neighbors For eah suh ommon neighbor x, there are at least 3 n ommon neighbors of u, v, x For eah suh ommon neighbor x there are at least 4 3)n ommon neighbors of u, v, x, x Continuing in this way for k times, and notiing that eah K k is ounted at most k! times in this way, we obtain that 4) n k Πk i= i + ) i) fe) n k k! k! k Let KG) denote the set of K k of G and let α = KG) /n k For a opy K KG), we have that ψk) = Consider first the ase where 4 α In this ase we get the trivial bound k! max e ET ) fe) k n k k k! + k ) k Π k i= i+) i) ψ = ψk) αn k k! k n k K KG) k ) ) ) k Π k i= i + ) i) + ) n k ) k For α whih is at most the rhs of 4, we onstrut the rational sequene B as in the proof of the previous setion Now B has ) k αn k = k KG) elements, whose sum is m Eah element of the sequene B is mapped to some K KG) and for eah K KG) there are k elements of B that are mapped to K, one for eah edge of K The first of these k elements in the sequene B is now the leading element with respet to K 7

8 Let B be the subsequene of B onsisting of the leading elements By definition of ψ, the sum of the elements of B is preisely ψ But B is some subsequene of KG) elements of the nondereasing sequene B, so in partiular, its sum is at least the sum of the first KG) = αn k elements of B Hene, ψ b l αn k l= Summarizing, we have that B is a nondereasing sequene of k αn k rationals, whose sum is m where m n / Eah element in the sequene is at least and k! n k Π k i= k! k n k at most and we wish to lower bound the minimum possible sum of i+) i) the first αn k elements of this sequene Let S = {s, s, } be any nondereasing sequene of k αn k rationals, whose sum is m, whose elements are between the two stated bounds, and for whih the sum of the first αn k elements is minimized As in the previous setion, we may assume that all the first αn k elements of S are equal So, we may assume that the first αn k elements of S are all equal to some value /xn k ), and that the average value of the remaining k )αn k elements of S is some value /yn k ), and reall that we have /yn k ) m = n k Π k i= k αn k i= k! i+) i) It follows that ) s i = αn k k xn k + )αn k yn k Using m n / and y Π k i= i+) i)/k!) we obtain from the last equation that α x + k ) )α Π k i + ) i)/k!) i= Hene, αn k s i = αn k n xnk k ) ) )α Π k i + ) i)/k!) i= i= Plugging in the rhs of 4 whih is an upper bound on α, we obtain s i αn k i= k ) ) ) k Π k i= i + ) i) + ) n k ) k By the minimality of S we have that ψ b i s i αn k i= αn k i= k ) ) ) k Π k i= i + ) i) + ) k ) k n as required 8

9 5 Improved lower bound for η3, ) In this setion we prove Theorem 5 Let > > / be fixed Let G be a graph with n verties and minimum degree δg) n By Corollary, it suffies to prove that ψ, the natural frational triangle paking of G, satisfies ) ) ψ n on ) 3 ) As shown in Setion 3, we may assume that for any edge u, v) EG), either du) = δg) or dv) = δg) We will also use 3), as well as the following notations from Setion 3: The number of edges of G is m where m n / T G) is the set of triangles of G and α = T G) /n 3 3 The ordering of EG) as a sequene e,, e m where fe i ) fe i+ ) for i =,, m 4 The non-dereasing sequene B = {b, b, } of 3αn 3 rationals onsisting of m bloks where blok j has fe j ) elements that are all equal to /fe j ) 5 The mapping M : B T G) whih bijetively maps the blok of B orresponding to e j to the fe j ) triangles ontaining e j 6 The notion of a leading element with respet to T T G) 7 The subsequene B onsisting of the leading elements, one for eah T T G) Reall that the sum of the elements of B is preisely ψ Let β and r be parameters satisfying 5) > β > β4β ), r = α 4 3 Let F E be F = {e m, e m,, e m βn +} Thus, F onsists of the βn edges having the lowest values of f The following is an immediate onsequene of the result of Goodman [6] mentioned in the introdution Lemma 5 Let G[F ] be the spanning subgraph of G onsisting of the edges of F Then G[F ] ontains at least rn 3 on 3 ) triangles where r = β4β )/3 Proof By Goodman s theorem, a graph with n verties and ρ n edges has ρρ ) n 3) on 3 ) triangles Our graph G[F ] has βn edges so we an use ρ = β o) and the lemma follows Let B F be the subsequene of B onsisting of the last βn bloks namely, the bloks orresponding to the elements of F ) By Lemma 5, we know that at least rn 3 on 3 ) of the leading elements fall in B F In other words, the subsequene B of the leading elements has B B F rn 3 on 3 ) while B B \ B F ) α r)n 3 + on 3 ) Reall that our goal is to lower bound the sum of the elements of B As B is non-dereasing, if we replae B with the subsequene B of the same ardinality αn 3 ) onsisting of the first α r)n 3 elements of B and the first rn 3 elements of B F then the sum of the elements of B is at most the sum of the elements of B up to the negligible error term on )) Let, therefore, Sn,, α, β) denote the set of all non-dereasing sequenes with the following properties for eah S S: S ontains 3αn 3 rational elements, partitioned into m bloks, where n m n / The overall sum of the sequene S is m 3 Eah element is between /n) and / )n) 4 The sum of the elements in the last βn bloks is βn 9

10 5 Eah S is oupied with a sub-sequene S onsisting of the first α r)n 3 elements and also with the first rn 3 elements of the subsequene of S onsisting of the final βn bloks 6 The value of S, denoted V als), is the sum of the elements of S Observe that B Sn,, α, β) and that V alb) is just the sum of the elements of B whih is what we want to lower bound Let, therefore, V aln,, α, β) = min V als) S Sn,,α,β) Observe that if S S satisfies V als) = V aln,, α, β), then as shown in Setion 3, we may assume that the first α r)n 3 elements of S whih are also the first elements of S ) are all equal to some value x/n where x / Likewise, the remaining rn 3 elements of S are all equal to some value z/n where / ) z x The elements of S starting with the element at loation α r)n 3 and ending at the last element of blok m βn have values whih are at least x/n and at most z/n Denote the number of suh elements by tn 3 The remaining α t)n 3 elements whih are also the last elements of S) have values whih are at least z/n and at most / )n) With this notation we have that V aln,, α, β) is of the form V aln,, α, β) = V als) = n α r)x + rz) but we must still ompute x and z whih minimize the rhs of the last equation, subjet to their onstraints Lemma 5 The solution to program P, α, β) of Figure is a lower bound for V aln,, α, β)/n P, α, β) : minimize α r)x + rz st ) α r)x β αz + )βz )rz, x z Fig The program P, α, β) whose solution is a lower bound for V al, α, β) Proof Let S be as in the previous paragraph, namely V aln,, α, β) = V als) = n α r)x + rz) The target value of P, α, β) then follows from the fat that V aln,, α, β)/n is of the form α r)x + rz Constraint is also obvious from the onstraints for x and z stated in the paragraph preeding the lemma To see the Constraint ) we observe the following Notie first that we must have 5 α r)x + tz β Indeed, the sum of the elements of S in the first m βn bloks is preisely m βn On the other hand by the disussion in the paragraph before the lemma, it is at least n 3 α r)x/n) + tn 3 )z/n) Dividing by n and using m n /, 5 follows Notie next that we must have 53) rz + α t) β Indeed, the sum of the elements of S in the last βn bloks is preisely βn On the other hand by the disussion in the paragraph before the lemma, it is at least 0

11 n 3 rz/n) + α t)n 3 / )n)) Dividing by n, 53) follows Now, onstraint ) is obtained by ombining inequalities 5 and 53) Let p, α, β) denote the solution for P, α, β) Let Corollary 53 η3, ) p, α) = max p, α, β) β satisfying 5) { min max p, α), 0 α 6 α, α } Proof Let G be any n-vertex graph with minimum degree n and with αn 3 triangles The fat that 0 α /6 is trivial Lemma 5 proves that for any β satisfying 5), p, α, β)n on ) is a lower bound for the natural frational triangle paking of G Thus, p, α)n on ) is a lower bound for the natural frational triangle paking of G The other two bounds in the statement of the orollary have already been proved in Setion 3 The bound α/)n on ) is the trivial lower bound proved in 3, and is useful when α is relatively large The bound α )n on ) is the other lower bound for the frational natural triangle paking of G proved in 33) and is useful when α is relatively small In fat, observe that the bound for η3, ) proved in Setion 3 is just the minimum over all α of the maximum of the two bounds α and α In order to lower bound p, α) it would be simpler to ompute p, α, β) at the partiular point β = α/ ) The reason for hoosing this partiular value of β will be made apparent later However, we must make sure that hoosing β = α/ ) does not violate 5) For this to hold, α must therefore satisfy This is equivalent to requiring 54) > α > α 4, r = 8 < α < 8α ) 4 ) α 3 Thus, by the definition of p, α) as a maximum over all plausible β of p, α, β), we obtain that p, α) p, α, α/ )) whenever α satisfies 54) For notational onveniene, define { p, α, α/ )) if q, α) = 8 < α < ) 4, 0 otherwise Thus, we an restate Corollary 53 as 55) η3, ) min 0 α 6 max { q, α), α, α } The reason for using q, α) instead of p, α) beomes apparent by the fat that the program P, α, β) is signifiantly simplified by plugging in β = α/ ) The following is an immediate orollary of Lemma 5

12 Q, α) : minimize α r)x + rz st ) α r)x α )rz, x z Fig The program Q, α) whose solution is q, α) Corollary 54 Let 8 < α < ) 4 and r = α 8α ) )/3 Then, q, α) is the solution to program Q, α) of Figure Analyzing Q, α) Observe that whenever Q, α) has a feasible solution, q, α) is always at least as large as α/ Indeed, the target funtion is α r)x + rz = αx + rz x) but in any feasible solution, z x and x / Similarly, q, α) is always at least as large as target funtion satisfies α To see this, notie that by Constraint ), the α r)x + rz α )rz + rz α sine z So, we now speify a valid range of α where Q, α) has a feasible solution, and where we an analytially ompute this solution, and we are guaranteed that in this range, q, α) dominates the three terms in the max expression of 55) We define two points g) and h) where g) < h)) as follows The point h) is the smallest point for whih we an take x = z = / as a feasible and hene optimal) solution For this to hold, onstraint ) must satisfy α r) α )r Solving the last inequality for α and reall that r = α 8α ) )/3) we have that 56) h) = ) ) In partiular, at the point α = h) and above it) we have that q, α) oinides with α/ The point g) is the largest point for whih we an take x = / and still have a feasible solution For this to hold, onstraint ) must satisfy α r) α )r ) Solving the last inequality for α we get that 57) g) = ) ) 3 Sine we have shown that q, a) dominates the two other terms in the max expression of 55) in the range [g), h)] we have, in partiular, that α if α g), 58) η3, ) min q, α) if g) α h), 0 α 6 α if α h)

13 Also notie from the above disussion that at point α = h) we have q, α) = α/ and at point α = g) we have q, α) = α Solving Q, α) in [g), h)] We an analytially ompute q, α) at the range α [g), h)] Let fz) = α )rz + rz By Constraint ) in Figure we have that α r)x + rz fz) Now, fz) is onave and, in addition, gets its minimum when z is as large as possible subjet to onstraint in Figure But onstraint ) shows that in order to make z as large as possible, x should be as small as possible Hene, the solution to Q, α) is when x = and z satisfies the quadrati equation α r) = α )rz Thus, z = ) r 4 a ) a r ) and q, α) = α r + r 4 ) a ) a r ) By taking seond derivative reall that r is also a funtion of α) we have that q, α), as a funtion of α, is onave in [g), h)] Thus, its minimum is obtained either at q, g)) or q, h)) Comparing the two values we obtain that the minimum of q, a) in [g), h)] is obtained at α = h) Thus, by 58) we have that η3, ) h) ) ) = 3 ) We end this setion by graphially illustrating the three funtions on the rhs of 58) for the ase = 3/4 In this ase we have h) = h3/4) = 5/6 38/ and g) = g3/4) = 3/6 5/ Figure 3 shows that in the range [g3/4), h3/4)] the funtion q3/4, α) is onave and dominates both α = 4α 3 and α = 3 8 4α The minimum in 58) is attained for α = h3/4) 6 Quality of lower bounds We start by analyzing the quality of the worst ase ratio between the lower bounds given in Corollary 53 namely, Theorem 3 in the ases k = 3, 4) and the onjetured values given in Conjetures for k = 3 and for k = 4 reall that the onjetured values are also upper bounds) Consider first the ase k = 3 As the onjetured value is /6 for 3/4 and / /4 when / 3/4, the worst ase ratio is min { inf < , min The left expression is a monotone dereasing funtion in /, 3/4] and the right expression is a monotone inreasing funtion in [3/4, ] Hene, = 3/4 minimizes both expression and the obtained ratio is 075 Consider next the ase k = 4 As the onjetured value is / for 4/5 and / /3 when /3 4/5, the worst ase ratio is min { inf 3 < , min 4 5 } }

14 Fig 3 The three funtions in the rhs of 58) in the ase = 3/4 By taking derivatives, it is not diffiult to see that the left expression is a monotone inreasing funtion in /3, 4/5] and the right expression is also monotone inreasing in [4/5, ] Both expressions are equal at = 4/5 Hene, the minimum is when = /3 whih yields the infimum of the left expression) and the obtained ratio is 03 We now analyze the quality of the worst ase ratio between the lower bounds given in Theorem 5 and the onjetured value given in Conjeture Here we need to ompute { } where x) = y) = min inf < 3 4 x), inf 3 4 < y) ) ) 3 ) ) ) 3 ) 6 Using derivatives, it is not diffiult though tedious) to see that x) is a monotone dereasing funtion in /, 3/4] in fat, in /, )) and that y) is a monotone inreasing funtion in [3/4, ) in fat, in /, )); see Figure 4 They are trivially) both equal at = 3/4 whih is, therefore, the minimum point of their upper envelope The value at this point, and hene the obtained ratio, is x3/4) = y3/4) = 0 38)/6 > 076 4

15 Fig 4 The dereasing funtion x) and the inreasing funtion y) shown in the range 05, ) 7 Upper bounds We start by giving a simple proof showing that the values stated in Conjeture are, in fat, upper bounds for ηk, ) Proposition 7 k k if k+ <, ) ηk, ) k k 0 if 0 if k k+, k Proof If k, then ηk, ) = 0 by Turán s Theorem If k+ then the upper bound k k is trivial as any n-regular graph annot have more than n // ) k edge-disjoint opies of Kk It remains to onsider the ase k k+ Start with the Turán graph T n, k ) whih has no opy of K k We may assume than k divides n and that n is an integer, by the asymptoti nature of the definition of ηk, ) Let H be any graph that has n/k ) verties and is regular of degree + k )n Embed a opy of H in eah vertex lass of T n, k ) The resulting graph G is regular of degree + k )n + k )n = n Now, eah opy of K k in G must ontain an edge from one of the k embedded opies of H Thus n ν k G) k )eh) = k ) + ) k ) k = n k ) k Observe that the proof of Proposition 7 atually also gives the same upper bound for η k, ) Interestingly, the following proposition shows that in order to prove Conjeture, it suffies to prove it for ηk, k+ ) Proposition 7 If ηk, k+ ) = k, then Conjeture holds 5

16 Proof Assume that indeed ηk, k+ ) = k Consider first the ase > k+ Let G be a graph with minimum degree n We may assume that n is divisible by k, by the asymptoti nature of the definition of ηk, ) By a theorem of Hajnal and Szemerédi [0], any graph with minimum degree at least n /k) ontains a K k -fator assuming n is divisible by k), namely n/k vertex-disjoint opies of K k We may therefore greedily remove from G a set of r pairwise edge-disjoint K k -fators, as long as n k )r n /k) In fat, we will only use r = n + k+ k After deleting r edge-disjoint K k -fators, we obtain a graph G with minimum degree k+ )n Assuming ηk, k+ ) = k, we have that G has n /k ) on ) edge-disjoint opies of K k Thus, G has at least r n k + n k on ) = k k n on ) edge-disjoint opies of K k Hene, ηk, ) /k k) We remain with the ase k < < k+ Let G be a graph with minimum degree n We first need to establish the following laim: in order to inrease the minimum degree of the graph by and remain with a simple graph), it suffies to add at most n/ + on) edges We prove this laim in the next paragraph Let A V G) denote the set of verties with degree n and let B = V G) \ A be the verties with higher degree The number of edges of the ut B, A) is therefore larger than n B ) B On the other hand, the sum of the degrees of the verties of A is n A It follows that ea) > n A n B ) B )/ Denoting A = αn and denoting by G [A] the omplement of the subgraph of G indued by A we have: eg [A]) ) ) A α ea) n α + α) α) on ) Simplifying the last expression we obtain eg [A]) α ) )n on ) The maximum degree of eg [A]) is not larger than the maximum degree of G whih is at most )n Summarizing, G [A] is a graph with αn verties, maximum degree at most )n, and its number of edges is at least α / )n on ) A simple appliation of Tutte s Theorem see, eg, []) gives that G [A] has a mathing M of size at least α /n on) edges The number of unmathed edges of G [A] is therefore at most αn M = α)n + on) For eah unmathed vertex v A, pik an arbitrary edge of G inident with v Denote the set of seleted edges by P Hene, P M is a set of edges whose addition to G inreases the minimum degree by But P M = P + M α)n + α /n + on) = n + on) Having proved our laim, we see that given a graph G with minimum degree n, we an add to it at most /k + ) )n n/ + on)) edges and obtain a 6

17 graph G with minimum degree at least /k + ))n By our assumption that ηk, k+ ) = k we have that G ontains at least n /k ) on ) edge-disjoint opies of K k At most eg ) eg) of these opies of K k ontain an edge that does not appear in G It follows that G has at least n k on ) ) n ) k + n + on) = k ) n on ) k edge-disjoint opies of K k Hene, ηk, ) k k, as required 8 Conluding remarks The proofs of Theorem 3 and Theorem 5 are algorithmi Moreover, given a graph with minimum degree at least n, one an find a K k -paking of size at least ηk, )n o n )) in polynomial time This follows diretly from the fat that Lemma the theorem of Haxell and Rödl []) is algorithmi, as they show that the optimal frational paking found via linear programming) an be onverted to an integral one with only a negligible additive loss, in polynomial time Although our main results onsider pakings with omplete graphs, they have a natural extension to arbitrary fixed graphs H Let ηh, ) be the infimum over all onstants γ suh that any graph G of order n and minimum degree at least n has ν H G) γn o n )) As proved in [6], if χh) = k and EH) = r, then r ν H G) k νk G) on ) Thus, one an use the lower bound in Theorem 3, multiplied by k /r, and obtain a lower bound for ηh, ) Likewise, the lower bound in Theorem 5, multiplied by 3/r, serves as a lower bound for ηh, ) where H is a 3-hromati graph REFERENCES [] B Bollobás Extremal Graph Theory Aademi Press, 978 [] P Dukes Rational deomposition of dense hypergraphs and some related eigenvalue estimates Linear Algebra and its Appliations, 4369): , 0 [3] P Erdős, R Faudree, R Gould, M Jaobson, and J Lehel Edge disjoint monohromati triangles in -olored graphs Disrete Mathematis, 3-3):35 4, 00 [4] P Erdos, AW Goodman, and L Pósa The representation of a graph by set intersetions Canad J Math, 8:06, 966 [5] D Fisher Lower bounds on the number of triangles in a graph Journal of Graph Theory, 34):505 5, 989 [6] AW Goodman On sets of aquaintanes and strangers at any party Amerian Mathematial Monthly, 669): , 959 [7] T Gustavsson Deompositions of large graphs and digraphs with high minimum degree PhD thesis, University of Stokholm, 99 [8] E Győri On the number of edge disjoint liques in graphs of given size Combinatoria, 3):3 43, 99 [9] E Győri and Z Tuza Deompositions of graphs into omplete subgraphs of given order Studia Sientiarum mathematiarum Hungaria, :35 30, 987 [0] A Hajnal and E Szemerédi Proof of a onjeture of Erdős In Combinatorial theory and its appliations, II Pro Colloq, Balatonfüred, 969), pages North-Holland, Amsterdam, 970 [] PE Haxell and V Rödl Integer and frational pakings in dense graphs Combinatoria, ):3 38, 00 [] I Holyer The NP-ompleteness of some edge-partition problems SIAM Journal on Computing, 04):73 77, 98 [3] P Keevash The existene of designs arxiv:403665, 04 [4] P Keevash and B Sudakov Paking triangles in a graph and its omplement Journal of Graph Theory, 473):03 6, 004 [5] TP Kirkman On a problem in ombinatoris Cambridge and Dublin Mathematial Journal, :9 04, 847 7

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