EXACT BOUNDS FOR JUDICIOUS PARTITIONS OF GRAPHS

Transcription

1 EXACT BOUNDS FOR JUDICIOUS PARTITIONS OF GRAPHS B. BOLLOBÁS1,3 AND A.D. SCOTT,3 Abstract. Edwards showed that every grah of size 1 has a biartite subgrah of size at least / + /8 + 1/64 1/8. We show that every grah of size 1 has a biartition in which the Edwards bound holds, and in addition each vertex class contains at ost /4 + /3 + 1/56 1/16 edges. This is exact for colete grahs of odd order, which we show are the only extreal grahs without isolated vertices. We also give results for artitions into ore than two classes. 1. Introduction Many classical robles in grah theory deand that a certain quantity be axiized or iniized. For instance, given a grah G, the Max Cut roble asks for the largest biartite subgrah of G. Our ai in this aer to consider robles in which several quantities ust be iniized or axiized siultaneously. Probles of this tye are in general ore difficult, since the quantities are not usually indeendent. As in [], by a judicious artitioning roble we ean a artitioning roble in which we require all vertex classes (or all airs of vertex classes, or all triles, and so on to satisfy inequalities siultaneously. 1 For the Max Cut roble, it is easy to see by considering rando artitions that every grah of size contains a biartite subgrah of size at least /. We can do a little better by considering artitions into two alost equal vertex classes: for instance, if G = n then in a artition into two equal classes we exect to have / + /(4n edges between the classes, so G contains a biartite grah with at least this any edges. Edwards [6], [7] roved the essentially best ossible result that every grah of order n and size contains a biartite subgrah of size at least ( Work for this article was artly coleted at the Institute for Advanced Study. 1

2 B. BOLLOBÁS1,3 AND A.D. SCOTT,3 Recently, siler roofs of the result of Edwards have been given by Erdős, Gyárfás and Kohayakawa [9] and Hofeister and Lefann [10]. (In fact, sile roofs can be read out of Lehel and Tuza [11] and Locke [1]. Alon [1] roved that there is soe c > 0 such that if / is a sufficiently large square then we can irove on (1 by c 1/4, while it is never ossible to irove on (1 by ore that O( 1/4. Further results on biartite subgrahs have also been given by Erdős, Faudree, Pach and Sencer [8]. Max Cut and the ore general Max k-cut, which asks for the axiu size of a k-artite subgrah, are NP-hard, and have been the subject of vigorous investigation in both cobinatorics and couter science. Given a grah G, the Max Cut roble is equivalent to the roble of iniizing e(v 1 + e(v over biartitions V (G = V 1 V. In this aer we shall study biartitions in which we ai to control the values of e(v 1 and e(v. In articular, we are interested in the roble of iniizing ax{e(v 1, e(v }, and ore generally, for k, of iniizing ax{e(v 1,..., e(v k } over artitions V (G = k i=1 V i. Thus rather than bounding the l 1 nor of the sequence (e(v i k i=1, we are bounding the l nor. Rando artitions give us less hel here than for Max Cut: although in a rando artition V (G = V 1 V we exect each of e(v 1 and e(v to have /4 edges, bounding both quantities siultaneously is uch harder. Even roving a bound such as (1 + o(15/16, for instance, is not at all straightforward. Indeed, the ossible resence of large degrees in the grah eans that a sile rando artitioning will not suffice (see []. It was roved in [] that every grah G has a artition into k sets, with at ost e(g /( k + 1 edges contained in any set. Note that this is best ossible, as seen by considering K k+1. However, this is soe way fro the /k that rando artitions would suggest. Indeed, for grahs with ore edges it is ossible to do uch better. A deterinistic artial artitioning cobined with artingale ethods was used in [] to rove that for k, every grah of size has a artition into k sets, each of which contains at ost k + (34/5 (log k /5 edges.

3 JUDICIOUS PARTITIONS 3 How uch can these bounds be iroved? We cannot exect to be able to do better than /k + c k, since any artition of Kkn+1 into k classes has at least ( n + 1 = 1 ( kn k 1 k k n = k + k 1 + O(1 k edges in soe class. Our ai in this aer is to show that a bound of for /k + c k can be guaranteed. Indeed, we give a bound that, surrisingly, for every value of k, is best ossible for infinitely any values of. We also show that, for any k, our bounds are exact for colete grahs of order nk + 1, for any ositive integer n, and that these are the only extreal grahs without isolated vertices. For biartitions, we do ore: we extend the result of Edwards by showing that there is a biartition that satisfies both the otial bound for ax{e(v 1, e(v } and the bound (1 of Edwards.. Biartitions We begin by roving a result for biartitions. We shall deterine the extreal grahs for this bound at the end of the section; a result for artitions into k sets is given in the next section. Let us note that for a ositive integer l, any artition of the grah K l+1 ust have at least ( l+1 edges in one class. Writing = e(kl+1 = ( l+1, we get ( l+1 = + l and 4 4 l 4 = , and so the bound ( in Theore 1 below is best ossible whenever is of for ( l+1. It is surrising that this bound can be achieved, since we are iniizing ore than one quantity siultaneously. As a bonus, we shall see that we can in addition deand that the bound (1 of Edwards is satisfied (it is equivalent to (3 below. Theore 1. Let G be a grah with edges. Then there is a artition V (G = V 1 V with ( e(v i

4 4 B. BOLLOBÁS1,3 AND A.D. SCOTT,3 for i = 1,, and (3 e(v 1, V Proof. Let us consider artitions V (G = V 1 V with e(v 1 e(v and (4 Γ(x V Γ(x V 1 for all x V 1, and (5 e(v 1, V Such artitions exist: let V (G = U 1 U be a artition of G with e(u 1, U axial. We ay assue e(u 1 e(u. Now Γ(x U 1 Γ(x U for every x U 1, or else we could ove x fro U 1 to U to get a artition with ore edges going between the two sets, and hence (4 is satisfied. Furtherore, since e(u 1, U is axial, we know fro (1 that (5 is satisfied. Let V (G = V 1 V be a artition of V (G with e(v 1 e(v that satisfies (4 and (5 with e(v 1 inial. If e(v 1 satisfies ( then we are done. Otherwise, suose (6 e(v 1 = 4 + α, so (7 α Suing (4 over all x V 1, it follows that e(v 1, V + α and so e(v = e(v 1 e(v 1, V 4 3α. Now let H = G[V 1 ] and ick v V (H with d H (v inial nonzero. Consider the artition V (G = W 1 W, where W 1 = V 1 \ v and W = V {v}. Clearly (W 1, W satisfies (4 and e(w 1 < e(v 1, so we ust have either e(w > e(w 1 or else (W 1, W does not satisfy (5. We clai that (W 1, W satisfies ( and (5.

5 JUDICIOUS PARTITIONS 5 We first rove that (W 1, W satisfies (5. Since e(w 1 = e(v 1 δ, where δ = δ(h = d H (v, we have e(w 1, W = Γ(x W x W 1 = Γ(x V + δ x W 1 Γ(x V 1 + δ x W 1 (8 = (e(v 1 δ + δ = + α. ( , by (7. We now rove (. It follows fro (9 that e(w = e(w 1 e(w 1, W ( ( 4 + α δ + α (10 = 4 3α + δ. Now e(h ( δ+1, so 4 + α ( δ + 1 and hence, rearranging, we get (11 δ + α If (W 1, W does not satisfy ( then it follows fro (6 and (10 that (1 in {α, δ 3α} > It follows fro (11 that in{α, δ 3α} in { α, The right hand side is axiized when } + α α. 4α + 1 = + α + 1 4,

6 6 B. BOLLOBÁS1,3 AND A.D. SCOTT,3 which gives α = However, this contradicts (1, and hence ( ust be satisfied. Let us note that we use the result of Edwards only to rove (3. If we are content with e(v 1, V / then the result follows directly fro the roof above without aeal to (1. It is easy to deterine the extreal grahs without isolated vertices for Theore 1. Keeing the notation of the roof, if we can do no better than equality in ( then we ust have equality throughout the roof. In articular, (V 1, V ust satisfy ( with equality and, defining (W 1, W as in the roof, (10, (11 and (1 are also satisfied with equality. Thus H ust be a colete grah, ossibly together with soe isolated vertices. Furtherore, (4 is satisfied with equality, so there are no isolated vertices in H, and every vertex in H ust have exactly H 1 neighbours in V. Now consider (W 1, W : every vertex in W 1 has W 1 = H 1 neighbours in W 1 and H neighbours in W. If any vertex v in W has ore neighbours in W than in W 1, then oving v fro W to W 1 yields a artition (V 1, V that satisfies the conditions of the theore strictly, unless v is adjacent to every vertex of W 1, in which case oving any other vertex fro W 1 to W will do. Thus (W, W 1 also satisfies (4 and (5 with equality and e(w e(w 1, so G[W ] is colete. Counting the nuber of edges between W 1 and W, we see that all edges between the ust be resent, so G is colete. It is easy to check that G ust be odd, so the only extreal grahs are colete grahs of odd order. 3. Partitions into k classes In this section we shall rove results for artitions of a grah into k vertex classes. Our ain ai is to rove that for integers k,, every grah of size has a artition into k vertex classes, each of which contains at ost k + k 1 k ( edges. Let us note that, for n 1 any artition of K nk+1 into k vertex classes has at least n + 1 vertices in soe class, and ( n ( kn + 1 = k 1 k k n.

7 JUDICIOUS PARTITIONS 7 Since ( n = k 4 1, it follows that the bound above is best ossible for colete grahs of order nk + 1. Our aroach for k > will be to choose one vertex class at a tie. Thus we shall begin by roving a lea about losided artitions. Theore. Let G be a grah with edges and let 0 1. Then there is a artition V (G = V 1 V with (13 e(v 1 + c(, and (14 e(v (1 + c(,, where (15 c(, = (1 ( Proof. We ay assue 0 < < 1. Let q = 1. Let us consider a artition V (G = U 1 U with qe(u 1 + e(u inial. We ay assue that e(u 1 e(u q, or else exchange and q (note that c( is syetric in and q. Now every v U 1 satisfies (16 q Γ(v U 1 Γ(v U, or else we could have oved v fro U 1 to U. Let us now choose a artition V (G = V 1 V that satisfies (16, that has e(v 1 e(v q and, subject to this, has e(v 1 inial. Suose that (V 1, V does not satisfy (13. Then let It follows fro (16 that e(v 1 = + α. e(v 1, V q ( + α = q + q α,

8 8 B. BOLLOBÁS1,3 AND A.D. SCOTT,3 and so e(v = e(v 1 e(v 1, V ( (1 q 1 + q α = q 1 + q α. As in the roof of Theore 1, we let H = G[V 1 ] and ick a vertex v of inial degree δ in H. Letting W 1 = V 1 \ v and W = V {v}, we obtain a artition (W 1, W which satisfies (16, so has with e(w 1 < e(w q. Now e(w 1, W = Γ(x W x W 1 = Γ(x V + δ x W q Γ(x V 1 + δ x W 1 = q (e(v 1 δ + δ = q e(v δ = q + q α + 1 δ. If e(w 1 > then suing (16 over W 1 gives e(w 1, W q and so e(w = e(w 1 e(w 1, W < (1 q = q, and thus both (13 and (14 are satisfied. If this is not the case then e(w 1 = M + α δ <. Now e(w = e(w 1 e(w 1, W ( ( + α δ q + q α + 1 δ = q α + 1 δ. If neither (V 1, V nor (W 1, W satisfies (13 and (14 then { (17 in α, 1 δ } ( α > q

9 JUDICIOUS PARTITIONS 9 As in the roof of Theore 1, we have (18 δ + α , and so { in α, 1 δ } α is at ost { α, 1 + α The latter exression is axiized when α = 1 + α and hence which contradicts (17. α = q ( , 4 } α. α It follows iediately by reeated alications of Theore that every grah G has a artition V (G = k i=1 V i, such that e(v i k + c k, where c k deends only on k. However, by being a little ore careful we can do uch better. Theore 3. Let G be a grah with edges. Then G has a vertex artition into k sets such that each set sans at ost ( (19 k + k k 4 1 edges. Proof. Let = ( kn+1, where n need not be an integer. Alying Theore with = 1/k, we get a artition V (G = V 1 V with (0 e(v 1 + c(, k and ( k 1 (1 e(v + c(,, k

10 10 B. BOLLOBÁS1,3 AND A.D. SCOTT,3 where Now since ( (1 c(, = ( = k k = = nk(nk ( nk we have Thus and e(v = nk, c(, = k 1 k n. e(v 1 + c(, k n(nk + 1 = + k 1 k k n ( n + 1 = = = = ( k 1 + c(, k (k 1 n(nk k 1 k k n (k 1 (nk n + 1 ( n(k Reeating this arguent, we ay divide G into k sets, each containing at ost ( n+1 edges. Now ( n + 1 n(n + 1 = n(n + 1 k k = k 1 k n.

11 JUDICIOUS PARTITIONS 11 However, = k n + kn, so ( n = 1 k and hence ( n + 1 e(v i k + k 1 k ( To investigate the extreal grahs for Theore 3, it is enough to know the extreal grahs for Theore. For = a/b, where (a, b = 1, and a ositive integer n, consider K bn+1. Let = e(k bn+1 = ( bn+1, so ( n = 1 b In any biartition V (K bn+1 = V 1 V, either ( an + 1 e(v 1 ( bn + 1 = + an b a b ( = + ( or e(v ( (b an + 1 ( bn + 1 = (1 = (1 + (1. (b an + a b ( Thus Theore is exact for grahs of order bn+1, and an easy calculation shows that these are the only colete grahs for which Theore is exact. Now suose that Theore is exact for G. Using the notation of the roof of Theore, we see that we ust have equality throughout. In articular, it follows fro (18 that H = G[V 1 ] ust be a colete grah. Furtherore, (16 ust be satisfied with equality. Hence ust be rational, say = a/b with (a, b = 1. Now (W 1, W ust.

12 1 B. BOLLOBÁS1,3 AND A.D. SCOTT,3 satisfy (14 with equality. If any vertex v in W is not adjacent to every vertex in W 1 then oving v to W 1 yields a biartition satisfying both (13 and (14 strictly. Thus the biartite grah between W 1 and W is colete. Since G[W 1 ] is also colete, a sile calculation shows that G[W ] ust also be colete and so G ust be colete. We have shown that the extreal grahs for Theore are exactly colete grahs of order nb + 1 with = a/b, where a, b and n are ositive integers. For Theore 3, our roof yields equality in (19 exactly when we have equality at each stage of the artitioning rocess. It follows that we have equality iff G = K kn+1 for soe ositive integer n. 4. Conclusion ( We have given bounds that are best ossible for grahs with = rk+1 edges, where k is the nuber of sets in our artitions and r is an arbitrary ositive integer. However, it should be ossible to do a bit better when is not of this for. The situation here is siilar to that of Max Cut: the bound (1 of Edwards is exact for infinitely any, but Alon [1] has shown that for certain values it can be iroved by c 1/4. It would be very interesting to know the best ossible iroveent of ( for every value of. Natural lower bounds can be obtained by considering unions of colete grahs. The aroach we have used here ight also be useful for siilar robles on hyergrahs. It was roved in [3] that every 3-unifor hyergrah with edges has a artition into k sets such that no set contains ore than k /7 (log k 1/ edges. It sees likely that a stronger result should hold. However, we have not yet been able to gain uch iroveent using our ethods. A lower bound /k 3 + O( 1/3 is given by considering colete 3- unifor hyergrahs. In general, judicious artitioning robles are uch harder for hyergrahs than for grahs, and for k > 3 very little is known. Thus even artial results are of interest: for instance it would be of great interest to deterine whether every r-unifor hyergrah with edges has a artition into k sets, each of which contains k + o( r edges. The best bound known so far is roved in [5], where it is shown that every r-unifor hyergrah with edges has a artition into k

13 sets, each of which contains at ost edges. JUDICIOUS PARTITIONS 13 r 8 log r k + c r r/(r+1 r 5. Addendu Soe tie after subitting this aer, we discovered that soe revious work had been done on this roble. T. D. Porter [13] showed that every grah with 1 edges has a biartition in which each class contains at ost /4+ /8 edges. More recently, Porter [14] showed that if k is a ower of then every grah G with 1 edges has a artition into k sets V 1,..., V k, each containing at ost /k + /k edges, such that k i=1 e(v i /k. Porter [15] has also shown that, for k, every grah with 1 edges has a artition into k sets with at ost /k + k edges contained in each set. Porter and Bin Yang [16] show that every grah with 1 edges has a biartition in which each class contains at ost /4 + /18 edges, and for k a ower of a artition into k sets, each of which contains at ost /k + /k edges. Stronger results follow iediately fro our Theores 1 and. In articular, note that for k a ower of, we can find a artition of a grah G into k sets, each of which satisfies (19, by reeated alication of Theore 1: given a artition into s sets, we biartition each set using Theore 1. A straightforward calculation shows each set in a artition obtained in this way satisfies (19; furtherore, it follows fro (3 that there are at ost /k edges with both ends in the sae vertex class. Finally, we note that Shahrokhi and Szekely [17] showed that the roble of finding a judicious biartition is NP-hard. References [1] N. Alon, Biartite subgrahs, Cobinatorica 16 (96, [] B. Bollobás and A.D. Scott, On judicious artitions, Periodica Matheatica Hungarica 6(1993, [3] B. Bollobás and A.D. Scott, On judicious artitions of hyergrahs, J. Cob. Theory, Ser. A 78 (97, [4] B. Bollobás and A.D. Scott, Judicious artitions of 3-unifor hyergrahs, Euroean J. Cob., to aear [5] B. Bollobás and A.D. Scott, Probles on judicious artitions, to aear [6] C.S. Edwards, Soe extreal roerties of biartite grahs Canadian J. Math. 5 (1973,

14 14 B. BOLLOBÁS1,3 AND A.D. SCOTT,3 [7] C.S. Edwards, An iroved lower bound for the nuber of edges in a largest biartite subgrah, in Proc., nd Czechoslovak Syosiu on Grah Theory, Prague 1975, [8] P. Erdős, R. Faudree, J. Pach and J. Sencer, How to ake a grah biartite, J. Cob. Theory Ser. B 45 (1988, [9] P. Erdős, A. Gyárfás and Y. Kohayakawa, The size of largest biartite subgrahs, Discrete Math. 177 (97, [10] T. Hofeister and H. Lefann, On k-artite subgrahs, to aear [11] J. Lehel and Zs. Tuza, Triangle-free artial grahs and edge-covering theores, Discrete Math. 39 (8, [1] S.C. Locke, Maxiu k-colorable subgrahs, J. Grah Theory 6 (8, [13] T.D. Porter, On a bottleneck conjecture of Erdős, Cobinatorica 1 (9, [14] T.D. Porter, Grah artitions, J. Cobin. Math. Cobin. Co. 15 (94, [15] T.D. Porter, Ars Cobinatoria, to aear [16] T.D. Porter, ersonal counication [17] F. Shahrokhi and L.A. Szekely, The colexity of the bottleneck grah biartition roble, J. Cobin. Math. Cobin. Co. 15 (94, Deartent of Matheatical Sciences, University of Mehis, Mehis TN 3815 Deartent of Matheatics, University College London, Gower Street, London WC1E 6BT, England 3 Trinity College, Cabridge CB 1TQ, England