Complete subgraphs n multpartte graphs FLORIAN PFENDER Unverstät Rostock, Insttut für Mathematk D-18057 Rostock, Germany Floran.Pfender@un-rostock.de Abstract Turán s Theorem states that every graph G of edge densty G / ) G > k contans a complete graph K k and descrbes the unque extremal graphs. We gve a smlar Theorem for l-partte graphs. For large l, we fnd the mnmal edge densty d k l, such that every l-partte graph whose parts have parwse edge densty greater than d k l contans a Kk. It turns out that d k l = k for large enough l. We also descrbe the structure of the extremal graphs. 1 Introducton and Notaton All graphs n ths note are smple and undrected, and we follow the notaton of [3]. In partcular, K k s the complete graph on k vertces, G stands for the number of vertces and G denotes the number of edges n G wth vertex set V G) and edge set EG). For a vertex x V G), let Nx) be the set of vertces adjacent to x, and let dx) := Nx) be the degree of the vertex. For sets X, Y V G), let G[X] be the graph on X nduced by G, EX) be the edge set of G[X] and EX, Y ) be the set of edges from X to Y. Let G be an l-partte graph on fnte non-empty ndependent sets V 1, V,... V l. For X V G), we wrte X := X V. For j, the densty between V and V j s defned as d j := dv, V j ) := G[V V j ]. V V j For a graph H wth H l, let d l H) be the mnmum number such that every l-partte graph wth mn d j > d l H) contans a copy of H. Clearly, d l H) s monotone decreasng n l. In [], Bondy et al. study the quantty d l H), and n partcular d 3 l := d lk 3 ),.e. the values for the complete graph on three vertces, the trangle. Ther man results about trangles can be wrtten as follows. Theorem 1. [] 1. d 3 3 = τ 0.618, the golden rato, and. d 3 ω exsts and d 3 ω = 1. Here, d 3 ω stands for the correspondng value for graphs wth a countably) nfnte number of fnte parts. They go on and show that d 3 4 0.51 and speculate that d3 l > 1 for all fnte l. We wll show that ths speculaton s false. In fact, d 3 l = 1 for l 1 as we wll prove n Secton 3. In Secton 4, we wll extend the man proof deas to show that d k l := d lk k ) = k for large enough l. In order to state our results, we need to defne classes Gl k of extremal graphs. We wll do ths properly n Secton. Our man result s the followng theorem. 1
Theorem. Let k, let l be large enough and let G = V 1 V... V l, E) be an l-partte graph, such that the parwse edge denstes dv, V j ) := G[V V j ] V V j k for j. k 1 Then G contans a K k or G s somorphc to a graph n G k l. Corollary 3. For l large enough, d k l = k. The bound on l one may get out of the proof s farly large, and we thnk that the true bound s much smaller. For trangles k = 3), we can gve a reasonable bound on l. We thnk that ths bound s not sharp, ether. We conjecture that l 5 turns out to be suffcent. Theorem 4. Let l 1 and let G = V 1 V... V l, E) be an l-partte graph, such that the parwse edge denstes dv, V j ) := G[V V j ] 1 for j. V V j Then G contans a trangle or G s somorphc to a graph n G 3 l. Corollary 5. d 3 1 = 1. Extremal graphs For l k 1)!, a graph G s n Ḡk l, f t can be constructed as follows. For a sketch, see the fgure below. Let {π 1, π,..., π )! } be the set of all permutatons of the set {1,..., k 1}. For 1 l and 1 s k 1, pck ntegers n s such that Let n π 1) n π )... n π ) for 1 k 1)!, n 1 = n =... = n for k 1)! < l, and n s > 0 for 1 l. s V G) = {, s, t) : 1 l, 1 s k 1, 1 t n s }, and EG) = {, s, t), s, t ) :, s s }. Fgure 1: A sketch of a member of Ḡ4 l, all edges between dfferent colors n dfferent parts exst.
Let Gl k be the class of graphs whch can be obtaned from graphs n Ḡl by deleton of some edges n {, s, k), s, k ) : s s 1 < k 1)!}. All graphs n Gl k are l-partte and Gk l contans graphs wth mn d j k e.g., we get d j = k for all j f all n s are equal). For k = 3, the densty condton s fulflled for all graphs n Ḡ3 l, and for all graphs n G3 l whch have d 1, 1. For k > 3, ths descrpton s not a full characterzaton of the extremal graphs n the problem, as for some choces of the n s, the resultng graphs wll have lower denstes than stated n the theorem. We would need some extra condtons on the n s to make sure that the graphs fulfll the densty condtons. 3 Theorem 4 trangles In ths secton we prove Theorem 4. We wll start wth a few useful lemmas and ths easy fact. Fact 6. Let G = V 1 V, E) be a bpartte graph on n vertces wth G 1 n, and let X be an ndependent set. Then X 1 X 1 n. Proof. There are at most n pars of vertces v 1 v wth v V. If 1 n of them are edges, then at most 1 n of them can be non-edges. An mportant lemma for the study of d 3 ω n [] s the followng. Lemma 7. [] Let G = V 1 V V 3 V 4, E) be a 4-partte graph wth V 1 = 1, such that the parwse edge denstes dv, V j ) > 1 for j. Then G contans a trangle. Wth the same proof one gets a slghtly stronger result whch we wll use n our proof. In most cases occurrng later, X wll be the neghborhood of a vertex, and the Lemma wll be used to bound the degree of the vertex. For the sake of exposton, we present a slghtly modfed verson of the proof here. Lemma 8. Let G = V 1 V V 3, E) be a 3-partte graph and X an ndependent set, such that the parwse edge denstes dv, V j ) 1 for j and X 1 V for 1 3, wth a strct nequalty for at least two of the sx nequaltes. Then G contans a trangle. Proof. In the followng, all ndces are computed modulo 3. For {1,, 3}, consder the 4-partte graph G[X, Y, X +1, Y +1 ]. For the dfferent choces of, we get the three nequaltes dx, Y +1 ) + dy, X +1 ) + dy, Y +1 ). Indeed, f we fx the number of edges between V and V +1 and the szes of X, Y, X +1, Y +1, the above sum s mnmzed f we mnmze the number of edges between Y and Y +1. As X Y +1 + Y X +1 1 V V +1 and dv, V +1 ) 1, the sum must be at least. As we have strct nequalty n at least two of the sx nequaltes n the statement of the lemma, at least one of the three sums s n fact greater than, and so 3 dx, Y 1 ) + dx, Y +1 ) + dy 1, Y +1 ) = =1 and thus for some {1,, 3}, 3 dx, Y +1 ) + dy, X +1 ) + dy, Y +1 ) > 6, =1 dx, Y 1 ) + dx, Y +1 ) + dy 1, Y +1 ) >. Pckng ndependently at random vertces x X, y Y 1, z Y +1, the expected number of edges n G[{x, y, z}] s dx, Y 1 ) + dx, Y +1 ) + dy 1, Y +1 ) >, and therefore G[X Y 1 Y +1 ] contans a trangle. 3
As a corollary from Fact 6 and Lemma 8 we get Corollary 9. For l 3, let G = V 1 V... V l, E) be a balanced l-partte graph on nl vertces wth edge denstes d j 1, whch does not contan a trangle. Then for every ndependent set X V G), X l+1)n. Proof. We may assume that X 1 X... X l. By Lemma 8, X 3 1 n and by Fact 6, X 1 + X 3 n. Now we are ready to prove Theorem 4. Proof of Theorem 4. Suppose that G contans no trangle. Wthout loss of generalty we may assume that each of the l 1 parts of G contans exactly n vertces, where n s a suffcently large even nteger. n Otherwse, multply each vertex n each part V by a factor of V, whch has no effect on the denstes or the membershp n Gl 3, and creates no trangles. For a vertex x, let d x) := Nx) V. For each edge xy EG), choose and j such that x V and y V j, and let sxy) := dx) d j x) + dy) d y). We have xy EG) sxy) = 1 x V G) y Nx) sxy) = x V G) dx) l d j x). The set Nx) s ndependent, so by Lemma 8, for fxed x at most two of the d j x) may be larger than n, and by Fact 6, d j x)d k x) 1 n for every vertex x V and j k. Thus, for fxed dx) n, the sum d j x) s maxmzed f n, f j = 1 and dx) n, n d j x) =, f j dx) n 1, dx) j n, f j = dx) n, and 0, otherwse, n whch case For fxed dx) < n, we have l d j x) = n + dx) n) n. j=1 j=1 l d j x) dx) < n + dx)) n = n + dx) n) n. j=1 Therefore, usng that dx) = G l ) n, 1 sxy) G dx) xy EG) x V G) = dx) n ln3 dx) dx) ln dx) n 3 ln dx) l )n n l 1. 4 dx) n dx) n) n )
We conclude that there s an edge xy EG) wth sxy) l )n n l 1. By symmetry, we may assume that x V 11 and y V 1. Note that Nx) and Ny) are dsjont as otherwse there would be a trangle. Let G := G[ 10 =1 V ]. Let X := Nx) V G ), Y := Ny) V G ), and Z := V G ) \ X Y ). Note that Z 11 n. By Lemma 8, at most two of the sets X and at most two of the sets Y are greater than n, so we assume n the followng that X n for 1 8 and Y n for 1 6. Further, we may assume that X 9 mn{ X 10, Y 7, Y 8 }. Let X X X Z and Y Y Y Z such that 1. X Y =,. X Y = V G ), 3. Y = max{ Y, n } for 1 8, and 4. X = max{ X, n } for 9 10. Let H := G EV 10, V 7 V 8 ). Let H H[X Y ] be the complete bpartte graph on X and Y, mnus the edges nsde the V and the edges between V 10 and V 7 V 8. We want to bound H from above. We have 8 n, for z Z 10, d H z) 10 n, for z Z 9, and 9 n, for z Z \ Z 9 Z 10 ), by Corollary 9. On the other hand, we have, usng that n X 7 + X 8 n, { 8 d H z) n, for z Z 9, and 7 n, for z Z \ Z 9. To see that d H z) 7 n for z Z 10, note that Z 10 Y 10 f Y 10 < 1, and thus d H z) 6 n + X 9. Therefore, takng nto account a possble double count of edges n the bpartte graph H [Z], we have Now, H H + n Z + 1 4 Z. H = 39 n + X 9 Y 10 + X 10 Y 9 1) + X 7 Y 8 + X 8 Y 7 ) + X 9 Y 7 + Y 8 ) + X 7 + X 8 ) Y 9. 3) For fxed X 9 n, 1) s maxmzed for mnmal X 10 X 9, and 3) s maxmzed for maxmal Y 7 + Y 8. For fxed Y 7 + Y 8, ) s maxmzed for maxmal Y 8 Y 7. Thus, 1)+)+3) s maxmzed for X 10 = Y 7 = X 9, n whch case 1) + ) + 3) = n. Ths shows that H 43 n, and thus H 43 n + Z n + 1 4 Z 43 n + 3 11 n. 5
On the other hand, by the densty condton, H 43 n, so EH ) \ EH) 3 11 n. In partcular, no vertex z can have large neghborhoods n both X and Y, as Nz) s an ndependent set and ths would force EH ) \ EH) to be large. To be more precse, let X := X \ X 10 and Ȳ := Y \ Y 10, then we have Nz) Ȳ n) Nz) X < 3 11 n, 4) as every vertex n Nz) X forces Nz) Ȳ ) \ V > Nz) Ȳ n mssng edges. Note that X, Ȳ 5n by Corollary 9. Let G := G V 10, and let X := {v V G ) : Nv) X > 1 X }, Y := {v V G ) : Nv) Ȳ > 1 Ȳ }, and Z := V G ) \ X Y ). The sets X and Y are dsjont by 4). As any two vertces n X or Y ) have a common neghbor, X and Y are ndependent sets. If z Z and Nz) Ȳ 6 5n, then d H z) Nz) Ȳ + Nz) X + Nz) Z 4) Nz) Ȳ + 3n + Z. 5) 11 Nz) Ȳ n) The last expresson s a convex functon n Nz) Ȳ and thus maxmzed on the boundary of the nterval [ 6 5 n, 5 n]. In the case Nz) Ȳ = 5 n, 5) gves For Nz) Ȳ = 6 5n, 5) gves d H z) 5 n + 46 363 n + 11 n <.81n. d H z) 6 5 n + 115 11 n + 11 n <.4n. We get the same upper bound wth a symmetrc argument for Nz) X 6 5n the symmetrc statement of 4) also holds). Fnally, f Nz) X + Nz) Y 1 5 n, then d H z) 1 5 n + 11 n <.6n. Every vertex z Y Z s ncdent to at least 1 X X 1 9n Ȳ Z ) n 10 11n edges n EH ) \ EH). So we have Y Z 3 11 11 10 n < 0.1n, and smlarly, X Z < 0.1n. Thus, Z < 0.4n. 6
Lke above, we may assume after possbly renumberng the sets) that X n for 1 7 and Y n for 1 5. Further, we may assume that X 8 mn{ X 9, Y 6, Y 7 } swtch Y s and Xs f necessary). Let H := G EV 9, V 6 V 7 ). By the densty condton, H 34 n. On the other hand, we can repeat the above arguments for H for H, and create a bpartte graph H on X X and Y Y wth d H z) 6 n for all z Z, and conclude that H 34 n 6.81) n Z + 1 4 Z 34 n 0.08n Z. Therefore, H = 34 n and Z =. Ths shows that d 3 l = d3 1 = 1. But more s true, G[ 8 V ] = H \ V 9 s a complete bpartte graph mnus the edges nsde the V, and we may assume that X 1 = X = X 3 = 1 n, as at most one of the X and at most one of the Y 1 8) may be greater than 1 n by the densty condton. For 9 k l, 1 8, 1 j 8 wth j, for every v V k, we have Nv) X Nv) Y j = 0 as otherwse there s a trangle. Thus, Nv) V 1 V V 3 ) 3 n wth equalty only for Nv) X = or Nv) Y =. Snce d k 1, equalty must hold for every v V k, showng that G s somorphc to a graph n Gl 3. 4 Theorem complete subgraphs Graphs whch have almost enough edges to force a K k ether contan a K k or have a structure very smlar to the Turán graph. Ths s descrbed by the followng theorem from [1], where a more general verson s credted to Erdös and Smonovts. Theorem 10. [1, Theorem VI.4.] Let k 3. Suppose a graph G contans no K k and G = 1 1 ) ) G k 1 + o1). ) Then G contans a k 1)-partte graph of mnmal degree 1 1 + o1) G as an nduced subgraph. Proof of Theorem. For the ease of readng and snce we are not tryng to mnmze the needed l, we wll use a number of varables l and c > 0 dependng on l. As l s chosen larger, the l grow wthout bound and the c approach 0. Let G be an l-partte graph wth V G) = V 1 V... V l wth denstes d j k, and suppose that G contans no K k. Wthout loss of generalty we may assume that each of the V contans exactly n vertces, where n s an nteger dvsble by k 1. We have G 1 1 k 1 1 ) ) G. l Let H be the k 1)-partte subgraph of G guaranteed by Theorem 10, wth parts V H) = X 1 X... X and Z := V G) \ V H). Further by Theorem 10, there s a c 1 > 0 dependng on l, so that Z c 1 G, and ths c 1 becomes arbtrarly small f l s chosen large enough. In partcular, Z c 1 n for at least 7
half the ndces 1 l. By the pgeon hole prncple, we can renumber the V and the X j, such that Z c 1 n and X 1 X... X for 1 l 1, where l 1 := l )!. ) For c = k 1)c 1, there s at most one ndex l 1 wth X 1 > 1 + c n, as otherwse there s a par V, V ) wth d 1 1 n k X j Xj j=1 1 < 1 k 1 + c k k 1 c k 1 + 4c 1 = k k 1. ) k ) ) 1 1 + c k So we may assume that ) ) 1 k 1 kc n X j 1 k 1 + c n for 1 l 1 1 and 1 j k 1. Ths mples that G[X j, Xj ] > X j Xj c 3 n c 1 for, j j, 1, l 1 1, 1 j, j k 1 and some c 3 > 0 wth c 3 0. For every v l 1 1 V, fnd a maxmum set P v of pars s, j s ) wth 1, 1) s, j s ) l 1, k 1), s s, j s j s, and Nv) X js s ) > c 4 n, + 4c 1 where c 4 := k c 3. If there s a vertex v wth P v = k 1, then we have a K k as follows. If we pck a vertex v s ndependently at random n each Nv) X js s ), then the probablty that v s v s s an edge s larger than c 4 c 3 = k 1, and therefore the expected total number of such edges s greater than c k 4 k 1 ) k > ) 1. Thus, there s a choce for the vs nducng a K n Nv). So we may assume that P v k for all v. For 1 l 1 1, assgn v Z to one set Y j X j, f there s no par, j) n P v. If there s more than one avalable set, arbtrarly pck one of them. Now we reorder the V and Y j agan to guarantee that Y 1... Y for 1 l, wth l := l 1 1 )!. In the followng, only consder ndces l. Note that for v Y j j, Nv) Y < c 4 + c 1 )n for all but at most k dfferent j, as Y j \ X j Z j. ) Let Ȳ Y be the set of all vertces v Y wth Nv) Y j ) < 1 1 + c 5 l n for some j, c 5 := c + c 4. Note that the sets Y \ Ȳ are ndependent, as the ntersecton of the neghborhoods of every two vertces n ths set contan a K k. Every vertex n v Ȳ j may have up to ) c 4 + c 1 )l k + 1) + k n neghbors n Y. But, at the same tme, v has at least Y 1 ) 1 k 1 + c 5 l n n > 1 3k l n 8
non-neghbors n some Y \ V j,. Then G[V 1... V l ] l j<j j<j ) l Y j j Y + Ȳ c 4 + c 1 )l k + 1) + k )n 1 3k l n ) l Y j j Y + Ȳ l n l j<j ) l k n, Y j j Y where equalty only holds f Ȳ = 0 for all, and Y j = ndex. Ths completes the proof of d k l = k ) c 4 + c 1 + k l 1 3k l n }{{} <0 for large enough l n for 1 j k 1 and all but at most one for large enough l. We are left to analyze the extremal graphs. After reorderng, we have Y j = n j and dy, Y j ) = 1 for 1 j, j k 1 and 1, k, f and j j. Let v V for some > k. Then Nv) k V kk ) n, as otherwse there s a K n Nv). On the other hand, equalty must hold for all vertces v V due to the densty condton. Therefore, Nv) k V = V \ Y j for some 1 j k 1. Defne Y j accordngly for all > k, and let Y j = Y j. Then V = Y j. For every permutaton π of the set {1,..., k 1}, there can be at most one set V wth Y π1) Y π)... Y π) and Y π1) > Y π). Otherwse, ths par = of sets would have densty smaller than k. Thus, all but at most k 1)! of the V have Y j for 1 j k 1. Therefore, all extremal graphs are n Gl k. 5 Open problems As mentoned above, the characterzaton of the extremal graphs s not complete for k > 3. We need to determne all parameters n s so that the resultng graphs n Ḡk l fulfll the densty condtons. The other obvous queston left open s a good bound on l dependng on k n Theorem, and the determnaton of the exact values of d k l for smaller l. In partcular, s t true that d3 5 = 1? Another nterestng open topc s the behavor of d l H) for non-complete H. Bondy et al. [] show that lm d lh) = χh) l χh) 1, but t should be possble to show wth smlar methods as n ths note that d l H) = χh) χh) 1 for large enough l dependng on H. References [1] B. Bollobás, Extremal Graph Theory, Academc Press London 1978). [] A. Bondy, J. Shen, S. Thomassé and C. Thomassen, Densty condtons for trangles n multpartte graphs, Combnatorca 6 006), 11 131. n 9
[3] R. Destel, Graph Theory, Sprnger-Verlag New York 1997). 10