On judicious bipartitions of graphs

Transcription

1 On judicious bipartitions of graphs Jie Ma Xingxing Yu Abstract For a positive integer m, let fm be the maximum value t such that any graph with m edges has a bipartite subgraph of size at least t, and let gm be the minimum value s such that for any graph G with m edges there exists a bipartition V G = V 1 V such that G has at most s edges with both incident vertices in V i. Alon proved that the limsup of fm m/ + m/8 tends to infinity as m tends to infinity, establishing a conjecture of Erdős. Bollobás and Scott proposed the following judicious version of Erdős conjecture: the limsup of m/4 + m/3 gm tends to infinity as m tends to infinity. In this paper, we confirm this conjecture. Moreover, we extend this conjecture to -partitions for all even integers. On the other hand, we generalize Alon s result to multi-partitions, which should be useful for generalizing the above Bollobás-Scott conjecture to -partitions for odd integers. School of Mathematical Sciences, University of Science and Technology of China, Hefei, Anhui, 3006, China. School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia , USA. Partially supported by NSF grants DMS and AST and NSA grant H

2 1 Introduction Let G be a graph and e EG. We will use V e to denote the set of the vertices of G incident with e. Let S, T V G. We write e G S := {e EG : V e S}, S, T G := {e EG : V e S = V e T }, and e G S, T := S, T G. If S = {v}, then we simply write e G v, T for e G {v}, T. For any integer and any -partition V 1, V,..., V of V G, let e G V 1, V,..., V = 1 i<j ev i, V j. When understood, the reference to G in the subscript will be dropped. Again let G be a graph. We write eg := EG. For any integer, let f G denote the maximum number of edges in a -partite subgraph of G. For any integers and m 1, let f m := min{f G : eg = m}. Let fg := f G and fm := f m. The problem for deciding fg, nown as the Max Cut Problem, is NP-hard, and there has been extensive wor on approximating fg, see [4, 14 16, 19]. On the other hand, it is easy to prove that any graph with m edges has a partition V 1, V with ev 1, V m/. Edwards [11, 1] improved this lower bound by showing fm m + 1 m 1 + 1/4 = m/ + m/8 + O1. 4 This is best possible, as K n+1 are extremal graphs. Erdős [13] conjectured that the limsup of fm m/ + m/8 tends to infinity as m tends to infinity. Alon [1] confirmed this conjecture with the following Theorem 1.1 Alon There exist absolute constants c > 0 and M > 0 such that for every even integer n > M, if m = n / then fm m/ + m/8 + cm 1/4. When m is sufficiently large, the function fm was determined exactly in a recursive formula by Alon and Halperin [3] and Bollobás and Scott [7] independently. Bollobás and Scott [5] considered problems in which one needs to find a partition of a given graph or hypergraph to optimize several quantities simultaneously. Such problems are called judicious partitioning problems, and we refer the reader to [7 9, 17] for more problems and references. Let G be a graph and a positive integer; we use g G to denote the minimal number t such that there exists a -partition V G = V 1... V satisfying ev i t for i = 1,...,. For any positive integers and m, let g m := max{g G : eg = m}. Let gg := g G and gm := g m. The problem of determining gg is nown as the Bottlenec Graph Bipartition Problem and is also NP-hard. The NP-hardness of determining gg can be derived from the NPhardness of Max Cut: taing a graph G consisting of two disjoint copies of graph H, then one can see that gg = eh fh. On the other hand, Bollobás and Scott [6] showed that for any positive integer m gm m m + 1/4 1/ = m/4 + m/3 + O1.

3 This bound is sharp, as K n+1 are extremal graphs. In fact, the result they proved is stronger, which states that this bound as well as the Edwards bound can be achieved simultaneously by some bipartite in every graph with m edges. Bollobás and Scott [7] also see [17] proposed the following judicious version of Erdős conjecture. Conjecture 1. Bollobás and Scott The limsup of tends to infinity as m tends to infinity. m/4 + m/3 gm The main goal of this paper is to prove the coming result which confirms Conjecture 1.. Theorem 1.3 There exist absolute constants d > 0 and N > 0 such that for every even integer n > N, if m = n / then gm m/4 + m/3 dm 1/4. We describe the main ideas of the proofs as follows. To establish an upper bound of gm, i.e., to find a bipartition such that each of its two parts contains a small number of edges, we start with a maximum cut, say V 1, V. In light of Theorem 1.1, this already guarantees that ev 1 + ev is relatively small. We then modify the bipartition by moving some vertices from the part with more edges say V 1 to the other part until V 1 has a small enough number of edges left, and eeping the number of edges in V not growing too much. Therefore, at each step we wish to choose a vertex in V 1 to move such that the number of edges lost in V 1 and the number of new edges added in V both are under control. To prove the existence of such a vertex say a good one, we show that there cannot be too many bad vertices. Similar approaches have been employed in [, 6, 10], while the challenges for us are to find suitable measures of goodness in all different cases, which are classified by the size of the maximum cut. Bollobás and Scott [8] also see [7] extended Edwards bound to -partitions of graphs and proved that for any graph G with m edges has a -partition V 1,..., V such that ev 1,..., V m + 1 m + 1/4 1/ + O, 1 with equality when G is the complete graph of order n + 1, where the O term is + 4 4/8. Bollobás and Scott [6] also showed that the vertex set of any graph G with m edges can be partitioned into V 1,..., V such that for i {1,,..., }, ev i m + 1 m + 1/4 1/. Recently, Xu and Yu [18] showed that a -partition V 1,..., V can be found that satisfies both inequalities 1 and simultaneously, generalizing an earlier bipartition result of Bollobás and Scott [6] as mentioned previously. It is natural to as whether Theorems 1.1 and 1.3 can be extended to -partitions. We show that the generalization of Theorem 1.3 to -partitions does hold for all even integers. 3

4 Theorem 1.4 Let > 0 be an even integer, and let d and m be the same as in Theorem 1.3. Then, g m m + 1 4d m m1/4 + O. We also prove the following generalization of Theorem 1.1, which we thin should be useful for extending Theorem 1.3 to -partitions for odd integers. Theorem 1.5 For any integer, there exist positive constants c = O1/ and N = O 3 such that for every even integer n > N, if m = n / then f m m + 1 m + cm 1/4. We organize this paper as follows. In Section, we prove Theorem 1.3 by establishing a sequence of lemmas. In Section 3, we first present the proof of Theorem 1.4, and then prove Theorem 1.5 by refining Alon s original proof of Theorem 1.1. In Section 4, we offer some concluding remars. Bipartitions In this section, we prove Theorem 1.3. Our proof is divided into several cases according to values of fg eg/. Since we will be using maximum bipartite subgraphs to find good judicious partitions, we need a result from [] which establishes a connection between fg and gg; a similar connection between f G and g G can be found in Bollobás and Scott [10]. Lemma.1 Alon, Bollobás, Krivelevich and Sudaov Let G be a graph with m edges and fg = m/+δ. If δ m/30 then there exists an absolute constant D such that, when m D there exists a bipartition V G = V 1 V satisfying max{ev 1, ev } m 4 m If δ m/30, then there exists a partition V G = V 1 V such that max{ev 1, ev } m 4 δ + 10δ m + 3 m. 4 The following easy consequence of Lemma.1 proves Theorem 1.3 for graphs G with eg = m and fg m/ m/10 4. Lemma. There exists an absolute constant M 1 > 0 such that the following holds. If G is a graph with m edges and fg = m/ + δ, m M 1, and δ m/10 4, then there exists a bipartition V G = V 1 V such that for i = 1,, ev i m 4 m

5 Proof. Let D be from Lemma.1. Let M 1 D be a sufficiently large constant here and later we do not express the constants explicitly to eep the presentation clear. Then by Lemma.1, we may assume that m/10 4 δ m/30, and that there exists a partition V G = V 1 V such that 4 holds. As the function hδ := δ/ + 10δ /m achieves its maximum at δ = m/40, we see that ev i m 4 + hm/ m m/4 m/ for each i = 1,. In view of Lemma., it suffices to prove Theorem 1.3 for graphs G with fg m/ m/10 4. We will find special vertices that we could use to modify existing bipartitions. This will be done in the next lemma and Lemma.5, using the approach described in the introduction. Lemma.3 Let G be a graph with m edges, and assume fg = m/ + δ, where δ m/10 4. Suppose V G = V 1 V is a partition such that ev, V 1 ev, V for every v V 1, and ev 1 m/4 δ/. Then there exists v V 1 such that ev, V 1 m/ + 3 δ and ev, V δ/mev, V 1. Proof. Let and T = {v V 1 : ev, V 1 > m/ + 3 δ} S = {v V 1 : ev, V > δ/m ev, V 1 }. We will show that v V 1 ev, V 1 > v S T ev, V 1, from which the existence of the desired vertex v follows. To this end, we bound v T ev, V 1 and v S ev, V 1. Since ev, V 1 ev, V for all v V 1, On the other hand, ev 1 = 1 ev, V 1 1 ev, V = 1 ev 1, V fg = m 4 + δ. v V 1 v V 1 ev 1 = v V 1 ev, V 1 v T ev, V 1 > m/ + 3 δ T, implying that T < ev 1 m/ + 3 δ m/ + δ m/ + 3 δ < m/ 3 δ, where the final inequality holds because δ m/10 4. Hence ev, V 1 ev 1 + et < ev T < ev v T m/ 3 δ. Since δ m/10 4, we have ev, V 1 < ev 1 + m 4 3 mδ/. 5 4 v T 5

6 Note that ev 1, V = v S ev, V + v V 1 S ev, V, so ev 1, V δ/m ev, V 1 + v S v V 1 S ev, V 1 = ev δ/m v S ev, V 1. Together with ev 1 m/4 δ/ and ev 1, V fg = m/ + δ, we get ev, V 1 < 1 m/δ ev1, V ev 1 1 mδ v S Combining 5 and 6, we have v S T ev, V 1 < ev 1 + m mδ/, implying that v V 1 ev, V 1 = ev 1 ev 1 + m 4 δ > ev 1 + m mδ/ > v S T ev, V 1, where the first inequality follows from ev 1 m/4 δ/ and the second inequality holds because δ m/10 4. So V 1 S T, completing the proof. The following result says that there exists an absolute constant C > 0 such that Theorem 1.3 holds for graphs G with m edges and m/8 + Cm 1/4 fg m/ m/10 4. Lemma.4 Let c be the absolute constant in Theorem 1.1, let d = c/4, and let C := 70+d. There exists an absolute constant M > 0 such that the following holds. If G is a graph with m edges, fg = m/ + δ, m M, and m/8 + Cm 1/4 δ m/10 4, then there exists a partition V G = V 1 V such that for i = 1,, ev i m/4 + m/3 dm 1/4. Proof. Let M > 0 be a sufficiently large constant. Let U 1, U be a maximum cut of G, i.e., eu 1, U = fg = m + δ. Without loss of generality we may assume eu 1 eu. Note that ev, U 1 ev, U for every v U 1 ; otherwise, eu 1 {v}, U {v} > eu 1, U = fg, a contradiction. Hence eu 1 = 1 ev, U 1 1 ev, U = 1 eu 1, U = m 4 + δ. v U 1 v U 1 We now define a process to move vertices from U 1 to U, using Lemma.3, such that in the end we get the desired bipartition. Initially, we set V1 0 := U 1, V 0 := U. Let V1 i V i denote the partition of V G after the ith iteration. If ev1 i m/4 δ/ + m/ + 3 δ/, set V 1 := V1 i and V := V i, and stop. If ev1 i > m/4 δ/ + m/ + 3 δ/ then by Lemma.3, there exists u i V1 i such that eu i, V1 i m/ + 3 δ and eu i, V i δ/m eu i, V1 i. Set V i+1 1 := V i 1 {u i}, V i+1 := V i {u i}, and repeat the above steps for V i+1 1, V i+1. 6

7 Note that, after i iterations for each i, we always have ev, V1 i ev, V i for every v V i 1 ; so Lemma.3 may be applied to V1 i, V i. Let V 1 = V1, V = V denote the final partition, after iterations. Then ev 1 m 4 δ + m/8 + 3 δ. Moreover, since V 1 is obtained from V 1 1 by moving u 1 to V 1, we have ev 1 > m 4 δ 1 m/ + 3 δ. Since ev 1 = eu 1 1 i=0 eu i, V1 i, we have 1 eu i, V1 i = eu 1 ev 1 < eu 1 m 4 + δ + 1 m/ + 3 δ. i=0 Hence, since eu = m eu 1, U eu 1 = m/ δ eu 1 and eu i, V i < δ/meu i, V1 i, we have 1 ev = eu + eu i, V i i=0 m δ eu δ/m eu i, V1 i m δ eu δ/m = m 4 δ m/ + m 4 δ m/ + i=0 eu 1 m 4 + δ + 1 δ + 0 δ/m eu 1 m 4 + δ + 1 δ + 0 δ/m δ m/ + δ = m 4 δ + m/8 + 3/ + 5 δ + 0 δ3/ m + 30 δ m. 3 m/ + δ 3 m/ + δ since eu 1 m/4 + δ/ Therefore, where max{ev 1, ev } m 4 + m/8 + hδ, hδ := δ + 3/ + 5 δ + 0 δ3/ m + 30 δ m. Since m M is sufficiently large, one can verify that hδ is a decreasing function in the domain m/8 + Cm 1/4 δ m/10 4. Therefore, max{ev 1, ev } m 4 + m/8 + h m/8 + Cm 1/4 < m 4 + m/3 C m1/4 + 70m 1/4 since m/8 + Cm 1/4 < m = m 4 + m/3 dm 1/4 since C = 70 + d, 7

8 completing the proof of this lemma. The next result is similar to Lemma.3, which will be used to prove Theorem 1.3 for graphs G with m edges and fg m/ m/8 + Cm 1/4. Lemma.5 Let c be the constant in Theorem 1.1 and d, C be the constants in Lemma.4. There exists an absolute constant M 3 > 0 such that the following holds. If G is a graph with m M 3 edges, fg = m/ + δ, m/8 + cm 1/4 δ m/8 + Cm 1/4, V G = V 1 V is a bipartition such that ev, V 1 ev, V for every v V 1, and ev 1 m/4 + m/3 dm 1/4 then there exists v V 1, such that ev, V 1 m/ + c 6 δ and ev, V 1 + c δ/m ev, V 1. 4 Proof. Let M 3 > 0 be a sufficiently large constant compared to c, d and C. And let { T = v V 1 : ev, V 1 > m/ + c } δ 6 and S = { v V 1 : ev, V > 1 + c } δ/m ev, V 1. 4 Since ev, V 1 ev, V for all v V 1, we have On the other hand, ev 1 1 ev 1, V fg ev 1 v T ev, V 1 > = m 4 + δ. c m/ + δ T. 6 Thus ev 1 T < m/ + c 6 δ m/ + δ < m/ c δ, δ 1 m/ + c 6 where the last inequality holds because m/8 + cm 1/4 δ m/8 + Cm 1/4 and m M 3 is large enough. From v T ev, V 1 ev 1 + et, we obtain that ev, V 1 ev v T Since ev 1, V = v S ev, V + v V 1 S ev, V, ev 1, V 1 + c δ/m ev, V v S c m/ δ ev1 + m 1 4 c mδ/. 7 4 v V 1 S 8 ev, V 1 = ev 1 + c δ/m ev, V 1. 4 v S

9 Therefore ev, V 1 4 m/δ ev1, V ev 1, c v S which, together with ev 1, V m/ + δ, ev 1 m/4 + m/3 dm 1/4 and δ m/8 + Cm 1/4, imply that ev, V 1 4 c C + d m/δ m 1/4. 8 v S Again, because ev 1 m/4 + m/3 dm 1/4, we have ev, V 1 = ev 1 ev 1 + m/4 + m/3 dm 1/4. 9 v V 1 When m M 3, in view of δ = Θ m, we see m/3 dm 1/4 > 4 c C + d m/δ m 1/4 c mδ/, 4 which, combining with 7, 8 and 9, imply that So we get V 1 S T as desired. v V 1 ev, V 1 > Now we are ready to prove Theorem 1.3. v S T ev, V 1. Proof of Theorem 1.3. Let c, M be the absolute constants in Theorem 1.1, and let M 1, M, M 3 be the absolute constants in Lemmas.,.4 and.5, respectively. Let d := c/4, M := max{m 1, M, M 3 }, and N max{m, M 1/ } be a sufficiently large constant. Let m = n /, where n N is even; so n M and m M i for i = 1,, 3, and hence we may apply Theorem 1.1 and Lemmas. and.4 and.5. Let G be a graph with m edges and fg = m/ + δ. By Theorem 1.1, Lemma. and.4, we may assume that m/8 + cm 1/4 δ m/8 + Cm 1/4. 10 Let V G = U 1 U, with eu 1, U = fg = m + δ and eu 1 eu. Then, as before, we see that ev, U 1 ev, U for every v U 1. We now describe a process similar to that in the proof of Lemma.4, to obtain the desired bipartition. Set V1 0 := U 1, V 0 := U, and let V G = V1 i V i iterations. be the partition obtained after i If ev1 i m/4 + m/3 dm 1/4, then set V 1 := V1 i and V := V i, and stop. 9

10 If ev1 i > m/4 + m/3 dm 1/4 then by Lemma.5, there exists a vertex u i V1 i such that eu i, V1 i m/ + c δ and eui, V i 1 + c δ/m eu i, V1 i. 6 4 Set V i+1 1 := V i 1 {u i}, V i+1 := V i {u i}, and repeat the above steps for V i+1 1, V i+1. Note that, in each iteration of the above procedure, we always have ev, V1 i ev, V i for every v V1 i ; thus Lemma.5 may be applied for V i 1, V i. Let V 1 = V1, V = V be the final partition, obtained after steps. Then ev 1 m/4 + m/3 dm 1/4, and ev 1 > m/4 + m/3 dm 1/4 c m/ + δ. 6 It suffices to show the upper bound of ev. Since ev, U 1 ev, U for every v U 1, Since ev 1 = eu 1 1 i=0 eu i, V1 i, we have i=0 eu 1 1 eu 1, U = m 4 + δ. 1 eu i, V1 i = eu 1 ev 1 < eu 1 m/4 m/3 + dm 1/4 + m/ + 6 c δ. Then, since ev = eu + 1 i=0 eu i, V i, eu i, V 1 i + c δ 4 m eu i, V1 i and eu 1 m/4 + δ/, we have ev m δ eu c δ 1 eu i, V1 i 4 m i=0 < m δ eu c δ/m eu 1 m 4 4 m/3 + dm 1/4 + m/ + c δ 6 = m 4 δ + 3 m/3 + dm 1/4 + c c δ + δ/m eu 1 m m/3 + dm 1/4 + c 6 m 4 δ + 3 m/3 + dm 1/4 + c 6 By 10, it holds that dm 1/4 + c/6 δ < m/3. Hence, δ + c 4 δ/m δ + 3 m/3 + dm 1/4 + c 6 ev m m/3 + dm 1/4 + hδ, δ. δ where c hδ := δ c δ c m δ3/ 10

11 is a decreasing function in the domain of 10, provided that m N / is sufficiently large. Therefore, we have ev m m/3 + dm 1/4 + h m/8 + cm 1/4 m m/3 + dm 1/4 m/8 c 4 m1/4 using m/8 + cm 1/4 < m = m 4 + m/3 dm 1/4 since d = c/4. This completes the proof of Theorem Partitions In this section, we first show Theorem 1.4, extending Theorem 1.3 to all multiple partitions with even parts. Proof of Theorem 1.4. Write := t for some integer t > 0 and let d, m be the same as in Theorem 1.3. Consider an arbitrary graph G with m edges. By Theorem 1.3, there exists a bipartition V G = V 1 V such that for i = 1,, e i := ev i m/4 + m/3 dm 1/4. Using the inequality 1 + x 1 + x for x 1, we have ei m/ / m/3 dm 1/4 /m/4 = m/ + O1. By the equation, each G[V i ] has a t-partition V1 i,..., V t i such that for j {1,,..., t}, evj i e i t + t 1 t e i + Ot m 1 4t + = m + 1 m 4d m1/4 + O. 8t + t 1 4t These two t-partitions together form a desired -partition of G. m d t m1/4 + Ot Next we generalize the proof of Alon in [1] to prove Theorem 1.5. We need the following lemma which appears in several articles, for example, as Lemma.1 in [1]. Lemma 3.1 Let G = V, E be an s-colorable graph with m edges. Then for any positive integer s, ts, f G s m, where ts, = s + i 1 s + j 1. 1 i<j In the coming proof of Theorem 1.5, we will be using Lemma 3.1 for ts,. Note that s ts, = s and ts, / = s 1. 11

12 Proof of Theorem 1.5. Fix, and let ɛ = /4, c = 8 ɛ, and N = 3 3. We do not attempt to optimize these constants. Let n be an even integer such that n N. Consider an arbitrary graph G with m = n / edges. Let s denote the unique integer satisfying n ɛ n + 1 < s n ɛ n Claim 1. We may assume that χg s + 1. For, suppose G is s-colorable. Then ts, f G m by Lemma 3.1 s = 1 m + 1 m s 1 m + 1 n n ɛ n + m + 1 n + ɛ n m + 1 This proves Claim 1. since m = n / and s n ɛ n since n N = 3 3 and ɛ = 1/16 m ɛm1/4 since m = n / and 1/4 1/8. Let H G such that χh = χg and H is vertex-critical. Then δh χg 1 s n ɛ n. Since eh n /, V H eh/δh n + ɛ n. Then there are at least n 4ɛ n color classes of size 1 in any proper coloring of H using χg colors. So there exists a complete subgraph R of G with V R := n r n 4ɛ n. Note that r 4ɛ n and Hence, n r er = er = n r + 1/ = n r + 1n rr O1 m ɛ 8n + O1, where the O1 term is 1/ 1/8. Let W := V G V R. Then ew + er, W = m er = rn r + n + r r. Claim. We may assume that ew n/8. Otherwise, assume ew n/8. By the equation 1, there exist -partitions W = i=1 W i and V R = i=1 R i such that ew 1,..., W ew + 1 ew + O ew + 1 n/ + O, 4 1

13 and er 1,..., R er + 1 er + 1 er + O 1 m 4ɛ n + O. For any permutation π [], we define eπ := i=1 ew i, R πi ; then eπ = 1! ew, R. π [] Thus there exists a permutation π [] such that eπ 1! ew, R/! = ew, R/. So ew 1 R π1,..., W R π =ew, R eπ + ew 1,..., W + er 1,..., R 1 ew, R + ew n/ + 4 er + 1 = 1 m m + m1/ O m + 1 m m1/4 + O. 1 m 4ɛ n + O Note that the O term here is + 4 4/8 + 1/ 1/8. Since m > n / > 3 3 /, we see that ew 1 R π1,..., W R π m + 1 m + cm 1/4. So the assertion of the theorem holds with the partition W 1 R π1,..., W R π, completing the proof of Claim. Let v 1, v,..., v n r be the vertices of R and let d i := Nv i W for 1 i n r, where d 1 d... d n r. Since R is complete, a balanced -partition of V R i.e., the sizes of parts differ by at most 1 gives f R. So let V R = i=1 R i be a -partition such that n r = R 1 R... R = n r, and let R 1 = {v 1,..., v n r er 1,..., R er + 1 er + O, }. Then where the O term is + 4 4/8. Let 1 i n r D := d i ew, R = n r n r, and D 0 := D D. Then, since ew + er, W = rn r + n/ + r r/, we have D 0 = n/ + r r/ ew. n r Since r 4ɛ n, n N and ew n/8, we conclude that min{d 0, 1 D 0 } 13 n 5n r.

14 Hence, D differs from an integer by at least n 5n r. Claim 3. ew, R... R 1 ew, R + n 5. To see this, let us consider two cases. If d n r D then d n r D+ n so Claim 3 follows as ew, R... R = d i i> n r 1 n r D + Otherwise, d n r D; then d n r D n 5n r ew, R 1 = d i i n r n r D n 5n r by integrality. Hence 5n r by integrality; = 1 1 ew, R + 5 n. n 1 5n r ew, R n 5 ; so ew, R... R 1 ew, R + n 5, completing the proof of Claim 3. Now, consider the -partition W R 1, R,..., R of V G. Since er 1,..., R 1 er+ 1 er + O and by Claim 3, We have ew R 1, R,..., R =er 1, R,..., R + ew, R... R er er + ew, R + n 5 + O m 1 ew m 4ɛ n + n 5 + O as er m ɛ 8m + O1 m m m 8 1 m 4ɛm1/4 + + O by Claim 5 m + 1 m + cm 1/4, where the last inequality holds because n N 3 3 and the O term is + 4 4/8. 4 Concluding remars We point out that Theorem 1.3 is best possible up to the constant d. To see this, let m = n +1. The lower bound on fm remains unchanged since the proof in [1] wors for values of m which differ from n / by a constant; so our proof also gives the same upper bound on gm. Let G be the vertex-disjoint union of K n and K, where n is odd, is even, and n = 1 1. Let m := eg = n +. An easy calculation shows that gg = n+1 + = m4 + m/3 m1/4 4 + O1. This shows that for m = n +1, gm m/4 + m/3 m 1/4 /8 + O1. 14

15 In our proof of Theorem 1.3, we choose d = c/4, where c is the constant from Theorem 1.1. The calculation in the end of the proof of Theorem 1.3 in fact only requires that d c+o1 d, where the O1 term may be made arbitrarily small when m is sufficient large. So one can show that for any ɛ > 0, there exists an integer N = Nɛ such that when n N, gm m 4 + c m/3 ɛ m 1/4. We do not now if one can get rid of the ɛ. Our proofs of Lemmas.3 and.4 may be modified to show that if G is graph with m edges, fg = m/+δ, and δ αm, then for sufficiently large m, there is a partition V G = V 1 V such that max{ev 1, ev } m 4 δ + m/8 + β δ + γ δ3/, m where α is an absolute constant and β, γ are constants depend on α only. When δ = Om t, where t < 3, this bound is better than Lemma.1, since m/8 dominates δ, δ 3/ / m while 3 m dominates 10δ /m in Lemma.1. Another conclusion we may draw from the proofs of Lemmas.3 and.4 is that: If fg m/+ m/8+α, where α = Θm t and 1 4 < t < 1, then gg m/4+ m/3+om 1/4 α/. We conclude our discussion with the following natural question: Is it also true that for every integer, the limsup of m + 1 m + 1/4 1/ g m tends to infinity when m tends to infinity? Theorem 1.4 has answered this question when is even. For the case that is odd, Theorem 1.5 and results of Bollobás and Scott in [10] seem to be relevant. Acnowledgment. We would lie to than two anonymous referees for their helpful comments and suggestions. We are grateful to a referee for bringing reference [3] to our attention and for arising a related question which enables us to discover Theorem 1.4. References [1] N. Alon, Bipartite subgraphs, Combinatorica [] N. Alon, B. Bollobás, M. Krivelevich and B. Sudaov, Maximum cuts and judicious partitions in graphs without short cycles, J. Combin. Theory Ser. B [3] N. Alon and E. Halperin, Bipartite subgraphs of integer weighted graphs, Discrete Math [4] P. Berman and M. Karpinsi, On some tighter inapproximability results, extended abstract Lecture Notes in Computer Science, Springer, Berlin, Heidelberg, [5] B. Bollobás and A. D. Scott, On judicious partitions, Period. Math. Hungar

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