Better bounds for k-partitions of graphs

Transcription

1 Better bounds for -partitions of graphs Baogang Xu School of Mathematics, Nanjing Normal University 1 Wenyuan Road, Yadong New District, Nanjing, 1006, China baogxu@njnu.edu.cn Xingxing Yu School of Mathematics Georgia Institute of Technology Atlanta, GA , USA yu@math.gatech.edu Abstract Let G be a graph with m edges, and let be a positive integer. We show that V G admits a -partition V 1,..., V such that ev i 1 m + 1 m + 1/ 1/ for i {1,,..., }, and ev 1,..., V 1 m + 1 m + 1/ + O, where evi denotes the number of edges with both ends in V i and ev 1,..., V = m i=1 ev i. This answers a problem of Bollobás and Scott [] in the affirmative. Moreover, +1 evi + ev j m + O for i {1,,..., }, which is close to being best possible and settles another problem of Bollobás and Scott []. Key words and phrases: Graph, graph partition, judicious partition AMS 000 Subject Classifications: 05C35, 05C75 Supported by NSFC Project Partially supported by NSA 1

2 1 Introduction The problem of finding a maximum bipartite subgraph in any graph is NP-complete. On the other hand, it is easy to show that every graph with m edges contains a bipartite subgraph with at least m/ edges. Edwards [, 5] improved this lower bound to m/ + hm/; where here and throughout hm = m The complete graphs K n+1 show that this bound is best possible. In [3] also see [], Bollobás and Scott extend Edwards bound to -partitions of graphs and prove that the vertex set of any graph with m edges can be partitioned into V 1,..., V such that ev 1,..., V := 1 i<j ev i, V j 1 m + 1 hm + O, 1.1 where ev i, V j is the number of edges with one end in V i and the other in V j. The inequality in 1.1 holds with equality when G is the complete graph of order n + 1, and the O term is determined in [3] as + /. In many situations, one often needs to find a partition of a graph that optimizes several quantities simultaneously; see [,6] for extensive discussions of such problems which are usually called Judicious Partitioning Problems. For example, Bollobás and Scott [1] showed that for any integer 1 and any graph G of size m, V G admits a partition V 1,..., V such that for i {1,,..., }, ev i m + 1 hm. 1. The complete graphs K n+1 are the only extremal graphs modulo isolated vertices. Another example is the following result of Porter [7]: If is a power of then every graph G with m edges has a partition of V G into V 1,..., V such that and for i {1,..., }, ev 1,..., V 1 m ev i m + m. Porter s result was improved by Bollobás and Scott [1] to ev i m/ + 1hm/. When =, Bollobás and Scott [1] obtained the following stronger result, where Nx denotes the neighborhood of the vertex x in a graph. Theorem 1.1 Bollobás and Scott [1] Let G be a graph with m edges. Then there is partition V 1, V of V G such that 1 Nx V Nx V 1 for all x V 1, ev i m + 1 hm for i = 1,, and

3 3 ev 1, V m + 1 hm. Condition 1 is essential in the proof of Theorem 1.1, as it allows one to move a vertex from V 1 to V so that the new partition also satisfies 1. The bounds in and 3 are individually tight; and the complete graphs K n+1 are the only extremal graphs modulo isolated vertices for Theorem 1.1. Bollobás and Scott [] ased whether this can be extended to 3. Problem 1. Bollobás and Scott [] Does any graph G of size m have a partition of V G into V 1,..., V that satisfies both 1.1 and 1.? A weaer version of Problem 1. is resolved in [] by the authors, which we shall use in this paper to deal with small graphs. Theorem 1.3 Xu and Yu [] Let G be a graph of size m, and let 1 be an integer. Then V G admits a partition V 1,..., V such that 1 for each i {1,... 1} and for every x V i, Nx j=i+1 V j i Nx V i, ev i 1 m + 1 hm for i {1,,..., }, and 3 ev 1,..., V 1 m + 1 hm. This result is stronger than the previous results when is restricted to powers of, and implies Theorem 1.1 when =. The main result of this paper is an affirmative answer to Problem 1.. Theorem 1. Let G be a graph of size m, and let 1 be an integer. Then V G admits a partition V 1,..., V such that 1 for each i {1,... 1} and for every x V i, Nx j=i+1 V j i Nx V i, ev i 1 m + 1 hm for i {1,,..., }, and 3 ev 1,..., V 1 m hm. The complete graphs K n+1 show that this is best possible, up to the O term. We have not made an effort to optimize the O term. Theorem 1. is closely related to another problem of Bollobás and Scott []. Problem 1.5 What is the largest c so that every graph G with m edges admits a partition of V G into V 1,..., V such that for each i {1,..., }, + 1 ev i + c ev j m. Bollobás and Scott note in [] that c = / if true would be best possible. In Section 3 we shall see that for large complete graphs we must have c < /. On the other hand, c can be arbitrarily close to / for sufficiently large graphs. More precisely, we shall prove the following in Section 3. 3

4 Corollary 1.6 Let G be a graph with m 1 edges, and let 1 be an integer. Then there is a partition of V G into V 1,..., V such that for i {1,..., }, + 1 Proof of Theorem 1. ev i + ev j m The main idea of the proof is the same as that used in []. That is, we wor with a partition V 1,..., V of V G that satisfies 1 of Theorem 1. called the property P in [], and the property that ev 1 ev i for all i {1,..., }. Such a partition may be produced by maximizing ev 1,..., V. If ev 1 1 m + 1 hm then we have the desired partition. Otherwise, we move a vertex from V 1 to V 1 := V G V 1, and inductively partition G[V 1 ] the subgraph of G induced by V 1 into 1 sets satisfying the property P 1. We need the following simple observation, proved in []. Lemma.1 Xu and Yu [] Let G be a graph and an integer, and let V 1,..., V be a partition of V G such that for each i {1,... 1} and for every x V i, Nx j=i+1 V j i Nx V i. Then 1 ev 1, V 1 1eV 1, and for any v V 1, ev 1 v, V 1 + v 1eV 1 Nv V 1. In above, V 1 v = V 1 \ {v} and V 1 + v = V 1 {v}. We also need the following, which shows that Theorem 1. holds for small graphs. Lemma. Theorem 1. holds when m 1/. Proof. Note that hm 1/ when m 1/. By Theorem 1.3, V G admits a partition V 1,..., V such that 1 and hold. In particular, for 1 i, ev i 1 m + 1 hm since m 1/ < 3. Therefore, we have ev i for i = 1,...,. Hence, ev 1,..., V m, and so 1 ev 1,..., V m + 1 hm 17 m 1 m + 1/ Consider gm := m 1 m + 1/ + + 1

5 as a function of m over the interval [0, 1/]. Differentiating with respect to m, we have g m = 1 1 m + 1/. So gm has a unique critical point at m = / in 0, 1/. Since g m > 0 on [0, 1/], we see that gm g / = Hence, 3 holds for V 1,..., V = 1 0. We now prove Theorem 1. by applying induction on. For = 1, Theorem 1. holds trivially, and for = it follows from Theorem 1.1. So we may assume 3. As induction hypothesis, we assume that the assertion of Theorem 1. holds when partitioning any graph into 1 sets. By Lemma., we may assume m 1.. Next we prove Claim 1. V G admits a partition V 1,..., V such that 1 and 3 of Theorem 1. hold for V 1,..., V, ev 1 max i {ev i }, and subject to the above conditions, ev 1 is minimal. To see this, let V 1,..., V be a partition of V G maximizing ev 1,..., V. Then by 1.1, the partition V 1,..., V satisfies 3 of Theorem 1.. Moreover, for any 1 i j and for any u V i, Nu V i Nu V j. So the partition V 1,..., V satisfies 1 of Theorem 1.. Without loss of generality, we may assume that So we have Claim 1. ev 1 max i {ev i}. Let ev 1 = m + α. If α 1 hm, we are done. So we may assume that α > 1 hm..5 Let H := G[V 1 ], δ the minimum nonzero degree in H, and v a vertex of H with degree δ. Let W 1 := V 1 v; so W 1 = V 1 + v and ew 1 = ev 1 δ = m/ + α δ. Moreover, for every u W 1, Nu W 1 Nu V 1 1 Nu V 1 1 Nu W

6 For convenience, write x := ew 1, W 1 and m := ew 1 = m ew 1 x. By of Lemma.1, m x 1eV 1 δ = 1 + α δ..7 Therefore, since ew 1 = ev 1 δ = m + α δ, we have m 1 m 1α + 1δ.. By induction hypothesis, W 1 admits a partition W,..., W such that for every i {,..., 1} and for every w W i, Nw W j i Nw W i,.9 j=i+1 where the neighborhood is taen in G[W 1 ], and such that and ew i m hm for i {,..., },.10 ew,..., W 1 m + 1 hm We wish to show that W 1,..., W gives the desired partition of V G for Theorem 1.. By.6 and.9, we have Claim. The partition W 1,..., W satisfies 1 of Theorem 1.. Next, we prove Claim 3. W 1,..., W satisfies 3 of Theorem 1.. From.11, we see that ew 1,..., W = ew 1, W 1 + ew,..., W x + 1 m + 1 hm We now prove that this lower bound on ew 1,..., W is at least 1 m hm. Let fx := x + 1 m hm 1 m 1 17 hm +. Recall that m = m ew 1 x and ew 1 = m/ + α δ. So m = m x m/ α + δ, and fx becomes x 1 m 1 α δ 1 m m x α + δ

7 Since x m ew 1 = 1m α + δ and because of.7, we have a x b,.1 where a := 1 m + α δ and b := 1m α + δ. Therefore, to prove Claim 3, it suffices to show that fx 0 for x [a, b]. Note that and f x = m x α + δ + 1 f x = 1 m x α + δ < 0. So f x = 0 has at most one solution in [a, b] at which if exists fx reaches a local maximum. Thus, to prove fx 0 for x [a, b], it suffices to show fa 0 and fb 0. First, we prove fb 0. Suppose, to the contrary, fb < 0. Then substituting x with b = 1m α + δ in fx and solving for α in fb < 0, we obtain α > 1 m + δ 1 m Recall that ew 1 = ev 1 δ = m + α δ. So m x + ew 1 m 1 + α δ + m + α δ by.7 = 1 m + 1α 1δ > 1 1 m 1δ + 1 m + δ 1 m + 1 > m m m + 1 > m. by.13 The final inequality holds because m 1/ by., the function lm := m m + 1 is increasing when m 1/, and l 1/ > 0. But this is a 1 contradiction. So fb 0. 7

8 Next we show fa 0. Substituting 1 m + α δ for a in fa, we get gα := fa 1 1 = 1 m + 1α δ m α δ 1 1 m m α + δ m 1α + δ + 1 = 1 α 1 m m α + 1δ + 1. Note that α must satisfy 1 m α + 1δ + 1 0, from which we deduce α β := 1 1 m δ By.5, α > θ := 1 m We now prove that gα 0 for α [θ, β]. Note that gβ = 1 β 1 m + 1/ Hence, gβ > m δ > 1 m 1 1 = 1 m 1 m + 1 > 0. m m The last inequality follows from. that m 1/, since the function lm := m m + 1 is an increasing function when m 1/ and l 1/ = 1/.

9 On the other hand, gθ = 1 m m m m δ + 1 = m m m δ > m m m + 1 = m + 1 Since > > 0. g α = m m m m α + 1δ + 1 1, g α = 0 has at most one solution in [θ, β]. Since g 1 1 α = 1 m α + 1δ < 0, gθ > 0 and gβ > 0 imply that gα > 0 for α [θ, β]. Therefore, fa > 0. This together with fb 0 shows that fx 0 for x [a, b], and hence the partition W 1, W,..., W satisfies 3 of Theorem 1., completing the proof of Claim 3. If the partition W 1,..., W also satisfies of Theorem 1., then by Claims and 3, W 1,..., W is the desired partition of V G for Theorem 1.. So we may assume that max ew i > m 1 i + 1 hm..1 The rest of the proof of Theorem 1. is exactly the same as that in Section 3 of [], starting from 3.9 in []. 9

10 3 Corollary 1.6 Before we prove Corollary 1.6, we show that for large complete graphs one needs c < / for the constant in Problem 1.5. Note that for any partition V 1,..., V of a graph G, we have + 1 ev i + ev j = + 1eV i + ev j = ev i + ev j j=1 = ev i + m ev 1,..., V. Let V 1,..., V be a -partition of K n+1 that maximizes ev 1,..., V. Then we must have V i V j { 1, 0, 1}. So we may choose notation so that V 1 = n + 1. Then ev 1 = 1 m + 1 hm and ev 1,..., V = 1 m + 1 hm +. Now let W 1,..., W be an arbitrary -partition of K n+1, and we may choose notation so that ew 1 = max{ew i }. Then, ew 1 ev 1 and ew 1,..., W ev 1,..., V. Hence, + 1 ew 1 + ew j j= = ew 1 + m ew 1,..., W ev 1 + m ev 1,..., V = m Therefore, the term c in Problem 1.5 must satisfy c < /. However, Corollary 1.6 shows that lim m c = /. Thus, in this sense, we have answered Problem 1.5. Proof of Corollary 1.6. By Theorem 1., there is a partition V 1,..., V of V G such that for each i {1,..., }, ev i m + 1 hm, and ev 1,..., V 1 m hm. 10

11 Then + 1 ev i + ev j = ev i + m ev 1,..., V m hm + m m + 1 hm 17 = m It is an interesting question to characterize the graphs G such that for every -partition V 1,..., V of V G there exists some i {1,,..., } such that + 1 ev i + ev j > m. References [1] B. Bollobás and A. D. Scott, Exact bounds for judicious partitions of graphs, Combinatorica [] B. Bollobás and A. D. Scott, Problems and results on judicious partitions, Random Structure and Algorithm [3] B. Bollobás and A. D. Scott, Better bounds for Max Cut, in Contemporary Comb, Bolyai Soc Math Stud 10, János Bolyai Math Soc, Budapest, 00, pp [] C. S. Edwards, Some extremal properties of bipartite graphs, Canadian J. math [5] C. S. Edwards, An improved lower bound for the number of edges in a largest bipartite subgraph, in Proc. nd Czechoslova Symposium on Graph Theory, Prague [6] A. Scott, Judicious partitions and related problems, in Surveys in Combinatorics, London Math. Soc. Lecture Note Ser., vol. 37, Cambridge Univ. Press, Cambridge, 005, pp [7] T. D. Porter, Graph partitions, J. Combin. Math. Combin. Comp [] B. Xu and X. Yu, Judicious -partitions of graphs, J. Combin. Theory Ser. B