Independence number of graphs with a prescribed number of cliques

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1 Idepedece umber of graphs with a prescribed umber of cliques Tom Bohma Dhruv Mubayi Abstract We cosider the followig problem posed by Erdős i Suppose that G is a -vertex graph where the umber of s-cliques i G is t. How small ca the idepedece umber of G be? Our mai result suggests that for fixed s, the smallest possible idepedece umber udergoes a trasitio at t = s/2+o1). I the case of triagles s = 3) our method yields the followig result which is sharp apart from costat factors ad geeralizes basic results i Ramsey theory: there exists c > 0 such that every -vertex graph with t triagles has idepedece umber at least { c mi log, log ) } 2/3. t 1/3 t 1/3 1 Itroductio A old problem i extremal graph theory due to Erdős [6] is to determie f, s, l), the miimum umber of s-cliques over all graphs o vertices Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA 15213, USA. tbohma@math.cmu.edu. Research supported i part by NSF grat DMS Departmet of Mathematics, Statistics, ad Computer Sciece, Uiversity of Illiois, Chicago, IL, USA. Research partially supported by NSF grat DMS mubayi@uic.edu 1

2 ad idepedece umber less tha l. This problem has bee extesively studied ad has led to may other iterestig questios. However, the rage of parameters that have bee ivestigated are cofied to the case of fixed l see, e.g., [5, 11, 10]). Here we cosider the problem whe s is fixed but l ca be a power of. It is more coveiet to phrase the problem i iverse form as follows. Suppose that s 2 is fixed ad G is a -vertex graph i which the umber of copies of K s is t, where 0 t s). How small ca the idepedece umber of G be? Whe s = 2 this questio is aswered by Turá s theorem. However, whe s 3, oe caot hope to obtai a precise aswer for this miimum for the full rage of t, sice whe t = 0 it is equivalet to determiig the Ramsey umbers rs, l) for fixed s ad large l. Ideed, our motivatio for studyig this parameter was to gai a better uderstadig of the Ramsey umber r4, l). I tryig to speculate about the growth rate of r4, l) a atural problem that arises is to determie the smallest possible idepedece umber of a graph where we have a specified umber of triagles the case s = 3 of our problem). Our first result below suggests that the behavior of this fuctio chages at t = s/2+o1). Theorem 1. Let s 2 be a fixed costat. If G is a graph o vertices with t copies of K s the c s 1 if t s/2 αg) c s ) 1 s s 2) if t s/2 t Remarks. 1. The s = 2 case of Theorem 1 is apart from costats) simply the boud give by Turá s Theorem. 2. The boud i the case of large t is essetially sharp from radom cosideratios. Ideed, if G is approximately radom ad has t copies of K s where t is sufficietly large) the G is roughly G,p with s p s 2) = 2

3 Θt). The we expect the idepedece umber of G, up to a logarithmic factor, to be 1/p. This is exactly the lower boud that we give i the large t case. 3. Although our proof gives some additioal log factors, we have chose to omit those for clarity of presetatio. For s = 3 we carry out the calculatios i our method i further detail ad provide more precise results below. Whe s = 3 Theorem 1 becomes { c 1 2 if t 3/2 αg) c if t 3/2 t 1/3 Here the bouds are sharp apart from logarithmic factors) i the etire rage of t due to the existece of Ramsey graphs with the desired idepedece umber [2, 3, 7, 9] for t < 3/2 ad approximately radom graphs for t > 3/2. However, our ext result sharpes both upper ad lower bouds cosiderably to obtai sharp results i order of magitude. Theorem 2. There exists c > 0 ad 0 such that if G is a graph o > 0 vertices with t triagles the c log if t 3/2 log αg) ) 2/3 c log if t 3/2 log. t 1/3 t 1/3 Both bouds are sharp apart from the costat c. The related questio of boudig the chromatic umber of a graph with a fixed umber of triagles was recetly addressed by Harris [8]. We ote i passig that the two problems have differet characters whe the umber of triagles is large. The extremal graph for the chromatic umber questio i this regime is simply a clique while the extremal graph for the idepedece umber questio is a direct product of a clique ad a Ramsey graph. Harris proof from [8] uses a difficult theorem of Johasso as its mai tool ad i fact his boud o chromatic umber together with a simple argumet ca be used to give aother proof of Theorem 2. Our proof uses oly the lower bouds o idepedece umber of triagle-free graphs; these are much easier results tha the correspodig upper bouds o chromatic umber. 3

4 2 Proof of Theorem 1 I the proof, we carry the costat c s through the calculatio i the iterest of clarity. All coditios that the costats must satisfy are listed as umbered equatios below. We make o attempt to optimize these costats. We go by iductio o s. Let d be the average degree of G. For a arbitrary vertex x let dx) deote the degree of x ad let tx) deote the umber of copies of K s that cotai x. Case 1. t s/2. Here we show that αg) > c s 1 slightly larger tha c s itself). where c s is a costat which will be If d c s) 1 s 2 the we have the desired boud by Turá s Theorem: αg) d c s s 2 = c s 1. So, we may assume d > c s) 1 s 2. Now cosider the radom variable X v = dv) tv) 2 where v is a vertex chose uiformly at radom. We have ] E [X v ] = E [dv) tv) 2 c s) 1 s 2 2 E [tv)] Note that we assume c s) 1 s 2 s s 2 1) 2 = ) c s) 1 s 2 s 2 > 0. c s < s 2. 1) Now we apply the iductive hypothesis i G[Nv)] where v is a vertex that achieves the above boud o X v. This graph has ) dv) > c s) 1 s 2 s 2 vertices ad at most tv) < dv) 2 4

5 copies of K. Therefore ) αg) αg[nv)]) c dv) 1 s 2 > c c s) 1 s 2 1 s 2 1. This gives the desired boud so log as ) c s c c s) 1 s 2 1 s 2. 2) Case 2. t > s/2. Here we radomly sparsify our graph ad apply Case 1. We may assume t < δ s where δ > 0 is a fuctio of the costat c s i the statemet of the Theorem. Set p = t 2/s 2 1+2/s ad cosider a set S of vertices chose at radom so that each vertex appears idepedetly with probability p. By stadard cocetratio results usig the upper t < δ s ) we have S > p 2 = /s t 2/s with high probability. Let T be the umber of copies of K s i G[S]. By Markov s iequality, we have T 2E[T ] = 2tp s = s 2 s+1 t with probability at least 1/2. Thus, there exists a vertex set S that satisfies both of these iequalities. So we ca apply Case 1 to coclude αg) αg[s]) c s S 1 c 2 s This gives the desired boud so log as 2 2s+1) 2 s) t s) c s < c s2 2s+1) s). 3). 5

6 3 Proof of Theorem 2 We begi with the lower bouds. The followig result will be used. It follows from the origial proof of Ajtai-Komlós-Szemerédi [1] see also Lemma i [4]). Theorem 3. There exists a absolute costat c such that the followig holds. Let G be a -vertex graph with average degree d ad t triagles. The αg) > c d log d 1 )) t 2 log. Now we cotiue with the lower boud proof of Theorem 2. Case 1. t 3/2 log ) 1/2. Let ɛ = 1/10. If d 1/4+ɛ, the usig Turás theorem we immediately obtai αg) /d+1) > c log with plety of room to spare, so assume that d > 1/4+ɛ. Next suppose that d > 7 log. Let v be a vertex chose uiformly at radom. Let dv) be the degree of v ad let tv) be the umber of triagles cotaiig v. The the expected value of dv) 2tv) is at least d 6t/ log ad sice αg) dv) 2tv) we are doe. Heceforth we may assume that 1/4+ɛ < d < 7 log. Now we apply Theorem 3 to obtai αg) c ) d d log > c log t where c > 0. Case 2. t > 3/2 log ) 1/2. Note that we ca assume t < δ 3 where δ > 0 is a fuctio of the costat c i the statemet of the Theorem. To be precise, δ ca be take to be solutio of the equatio c δ 1/3 log 1 ) 2/3 δ 1/3 = 1. Set p = 1 4 ) t 2/3 log ) 1/3 < 1. t 1/3 6

7 Choose a vertex subset S of V G) by pickig each vertex idepedetly with probability p. The, by stadard large deviatio estimates for p large ad stadard Poisso approximatio estimates for p costat, with high probability S p/2 := N. Ad by Markov s iequality, with probability at least 1/2, the umber of triagles T i S satisfies T < 2p 3 t = 1 ) 3 log ) 1/3 < N 3/2 32 t t 1/3 log N. Thus, we ca choose a set S that satisfies both of these iequalities ad apply the result we have already proved i Case 1 to obtai the lower boud αg) αg[s]) c N log N c 2 log/t 1/3 )) 1/3 2 log/t 1/3 )) = c 2 8t 2/3 t 1/3 ) log t 1/3 ) 2/3. Note that the last iequality holds for δ, ad hece c sufficietly small. Now we will exhibit costructios showig that these bouds are sharp. I the rage t 3/2 log ) 1/2 we proceed as follows. Choose a so that a 3) = t ad let G be a triagle-free graph o a vertices with idepedece umber at most c a) log a) this exists due to [9]). Note that the upper boud o t implies that a < 2 log = o). Let G be the graph that is a disjoit uio of G ad K a. The G has vertices, t triagles, ad αg) = αg ) + 1. Next we cosider the rage t 3/2 log ) 1/2. We may assume t < δ 3 for some δ > 0. Defie λ > 0 by the equatio t = 3/2 λ 3/2 log t 1/3 ad assume for simplicity of otatio that λ is a iteger. Let N = /λ ad agai assume that N is a iteger. Let G be a triagle-free graph o N vertices ad idepedece umber at most c N log N. Note that G has ΘN 3/2 log N) edges. Let G be the lexicographic product of G with K λ. I other words, replace each vertex v of G with a clique S v of size λ ad for x S v, we have N G x) = S v \ {v}) w NG v)s w. The G has λn = 7

8 vertices ad the umber of triagles i G is t = Θλ 3 EG ) ) = Θt). Fially, αg) = αg ) c N log N < c t 1/3 log/t1/3 )) 2/3. We have produced a costructio where the umber of triagles is t = Θt) ad it is a easy matter to produce oe where the umber of triagles is exactly t ad the idepedece umber has the same order of magitude. Ackowledgmet. The authors thak David Harris for poitig out that the results i [8] ca be adapted to yield aother proof of Theorem 2. The The first author thaks Gregory ad Nia Goosses. Refereces [1] M. Ajtai, J. Komlós, ad E. Szemerédi. A ote o Ramsey umbers. J. Combi. Theory Ser. A, 293): [2] T. Bohma, The triagle-free process, Advaces i Mathematics ) [3] T. Bohma ad P. Keevash, Dyamic cocetratio of the triagle-free process. arxiv: [4] B. Bollobás. Radom graphs, volume 73 of Cambridge Studies i Advaced Mathematics. Cambridge Uiversity Press, Cambridge, secod editio, [5] S. Das, H. Huag, J. Ma, H. Naves, B. Sudakov, A problem of Erds o the miimum umber of k-cliques. Eglish summary) J. Combi. Theory Ser. B ), o. 3, [6] P. Erdős, O the umber of complete subgraphs cotaied i certai graphs, Magyar Tud. Akad. Mat. Kutat It. Kzl [7] G. Fiz Potiveros, S. Griffiths ad R. Morris, The triagle-free process ad R3,k). arxiv: [8] D. Harris, Some results o chromatic umber as a fuctio of triagle cout. arxiv:

9 [9] J.H. Kim, The Ramsey umber R3, t) has order of magitude t 2 / log t, Radom Structures Algorithms ), [10] V. Nikiforov, O the miimum umber of k-cliques i graphs with restricted idepedece umber. Combi. Probab. Comput ), o. 4, [11] O. Pikhurko, E. Vaugha, Miimum umber of k-cliques i graphs with bouded idepedece umber, Combi. Probab. Comput ), o. 6,