Balanced bipartitions of graphs

Transcription

1 Dedicated to Professor Feng Tian on the occasion of his 70th birthday Balanced bipartitions of graphs Baogang Xu School of Mathematical Science, Nanjing Normal University baogxu@njnu.edu.cn or baogxu@hotmail.com 1 of 38 Supported by NSFC Joint work partially with Yan and Yu, and partially with Fan, Yu and Zhou)

2 Abstract A balanced bipartition of a graph G is a bipartition V 1 and V 2 of V (G) such that 1 V 1 V 2 1. In this talk, we will present (1) some results related to some problems and conjectures of Bollobás and Scott on balanced bipartition of graphs (this is a joint work with J. Yan and X. Yu), and (2) some results on a folklore conjecture concerning the balanced bipartition of plane graphs (this is a joint work with G. Fan, X. Yu and C. Zhou). 2 of 38

3 1 Introduction 2 A lower bound and an upper bound 3 Balanced judicious bipartition 4 Balanced bipartition of plane graphs 3 of 38 5 Some Open Problems

4 1 Introduction Let G be a graph, and let k 2 be an integer. A k-partition of G is a partition of V (G) into k pairwise disjoint nonempty subsets V 1,..., V k. When k = 2, we simply call a 2-partition as a bipartition. Usually, we ask some partitions with certain properties. For instance, we may ask the number of edges with ends in distinct subsets to be maximized. When k = 2, this problem is called the Maximum Bipartite Subgraph Problem (it is the unweighted version of the famous 4 of 38 Max Cut Problem).

5 Let V 1,..., V k be a k-partition of V (G). If V i V j 1 for every pair 1 i, j k, then this k-partition is called a balanced k-partition. A balanced 2-partition is also called a balanced bipartition. We use e(v i, V j ) to denote the number of edges with one end in V i and the other end in V j, and use e(v i ) to denote the number of edges 5 of 38 with both ends in V i. Let e(v 1,..., V k ) = 1 i<j k e(v i, V j ).

6 Edwards [7, 8] showed that every graph with m edges admits a bipartition V 1, V 2 such that e(v 1, V 2 ) m/ m ( ). This was generalized by Bollobás and Scott [2] as below. Theorem 1.1 (Bollobás and Scott [2]) Let G be a graph with m edges. Then there is partition V 1, V 2 of V (G) such that (1) e(v i ) m m ( ) for i = 1, 2, and 6 of 38 (2) e(v 1, V 2 ) m m ( ). The bounds in (2) and (3) are (individually) tight; and the complete graphs K 2n+1 are the only extremal graphs (modulo isolated vertices) for Theorem 1.1.

7 What can we say on balanced partitions, particularly on balanced bipartitions? In 2002, Bollobás and Scott asked the following problems [3]. Given a graph G, find a balanced partition V 1,..., V k of V (G) that minimizes max{e(v 1 ),..., e(v k )}. 7 of 38 For a graph G of size m, can we find a lower/upper bound for balanced bipartitions similar to that of Edwards bound?

8 In particular, Bollobás and Scott [3] made the following Conjecture 1 Every graph with m edges and minimum degree at least 2 admits a balanced bipartition V 1, V 2 such that max{e(v 1 ), e(v 2 )} m/3. The complete graph K 3 shows that the bound m/3 is sharp, and also the graphs obtained from disjoint triangles by identifying one vertex 8 of 38 from each triangle. The star K 1,n shows that the requirement on minimum degree is necessary.

9 Bollobás and Scott [5] proved Conjecture 1 for regular graphs. In fact, they proved that almost every regular graph with m edges admits a balanced bipartition V 1, V 2 such that max{e(v 1 ), e(v 2 )} < m/4. In [16], Yan and Xu proved that every graph G with m edges and (G) δ(g) 1 admits a balanced bipartition V 1, V 2 such that max{e(v 1 ), e(v 2 )} < m/4 + O(n). Xu, Yan and Yu [14] proved Conjecture 1 for graphs G with (G) 7 5 δ(g). It should be mentioned that the proving technique used in [16, 14] is the same as that used in [5]. They all showed that a balanced bipartition V 1 and V 2 with the largest e(v 1, V 2 ) has the required property. 9 of 38

10 We will present two sharp bounds on the maximum/minimum balanced bipartitions, some new progress on Conjecture 1, and some results on a problem concerning the minimum balanced bipartition of plane graphs. In contrast to the problem of finding a balanced bipartition that maximizes the number of edges with ends in distinct subsets, we consider the balanced bipartition of plane graphs that minimizes the number of edges with ends in distinct subsets. There is a folklore conjecture on this problem. 10 of 38 Conjecture 2 Every plane graph of order n has a balanced bipartition V 1, V 2 such that e(v 1, V 2 ) n. Finally, we end this talk with some unsolved problems.

11 2 A lower bound and an upper bound on balanced biparttitons Edwards bound asserts that every graph with m edges admits a bipartition V 1, V 2 such that e(v 1, V 2 ) m/ m ( ). Note that Edwards bound is a function of a unique variable m. It seems difficult to get such a bound for balanced bipartitions. We get a lower bound of maximum balanced bipartition problem as a function of m and the matching number. 11 of 38

12 Theorem 2.1 (Xu, Yan and Yu, DM 2010 inpress) Let G be a graph with m edges, and let M be a maximum matching of G. Then G admits a balanced bipartition V 1, V 2 such that e(v 1, V 2 ) (m + M )/2. It is easy to see that the bound in Theorem 2.1 is best possible for complete graphs. Since a maximum matching can be found in polynomial time [6], our proof implies a polynomial time algorithm for finding a balanced bipartition as described in Theorem of 38

13 Proof. Let M = {e 1, e 2,..., e r } be a maximum matching of G, and let V 1, V 2 be a balanced bipartition of V (G) such that (1) for each i {1, 2,..., r}, e i has one end in V 1 and the other in V 2, and 13 of 38 (2) subject to (1), e(v 1, V 2 ) is maximum.

14 Let V 1 = {u 1, u 2,..., u s } and V 2 = {v 1, v 2,..., v t }, such that s t and e i = u i v i for i = 1, 2,..., r. We can show that for i = 1,..., r, N(u i ) V 2 + N(v i ) V 1 N(u i ) V 1 + N(v i ) V and for i = r + 1,..., t, N(u i ) V 2 + N(v i ) V 1 N(u i ) V 1 + N(v i ) V of 38 Moreover, if s = t + 1 then N(u s ) V 2 N(u s ) V 1.

15 Suppose s = t. Then t 2m = ( N(u i ) V 2 + N(v i ) V 1 ) i=1 + 2 t ( N(u i ) V 1 + N(v i ) V 2 ) i=1 t N(u i ) V i=1 = 4e(V 1, V 2 ) 2r. Hence, e(v 1, V 2 ) (m + M )/2. t N(v i ) V 1 ) 2r i=1 15 of 38

16 Now assume s = t + 1. Then s s 2m = N(u i ) V 2 + N(u i ) V 1 i=1 + 2 t N(v i ) V 1 + i=1 i=1 s N(u i ) V i=1 = 4e(V 1, V 2 ) 2r. t N(v i ) V 2 i=1 t N(v i ) V 1 2r i=1 Again, e(v 1, V 2 ) (m + M )/2 as required. 16 of 38

17 With the similar arguments, we also have an upper bound of minimum balanced bipartition problem as below. Theorem 2.2 (Fan, Xu, Yu and Zhou, 2010 submitted) Let G be a graph with n vertices and m edges, and let M be a maximum matching of G c. Then G admits a balanced bipartition V 1, V 2 such that e(v 1, V 2 ) 1 2 (m + n 2 M ). The questions of finding the bounds of the form f(m) on balanced bipartitions are still open. 17 of 38

18 3 Balanced judicious bipartition In this part, we present some results concerning with Conjecture 1 and its related problems. A matching M of G is called a symmetric matching if for each edge uv M, u and v have the same degree in G. Theorem 3.1 (Xu, Yan and Yu, DM 2010 inpress) Let G be a graph with n vertices and m edges, and let M be a symmetric matching in G of maximum cardinality. Then, G admits a balanced bipartition V 1, V 2 such that e(v 1, V 2 ) (m + M )/2 and 18 of 38 (1) max{e(v 1 ), e(v 2 )} m M + (G) δ(g) 4 if n is even; (2) max{e(v 1 ), e(v 2 )} m M + (G) 4 if n is odd.

19 This result is best possible for K 1,n 1, n 2. Since a maximum matching can be found in polynomial time [6], a maximum symmetric matching can also be found in polynomial time (by considering the subgraph induced by 19 of 38 the edges whose ends have the same degree). Our proof of Theorem 3.1 implies a polynomial time algorithm for finding such a partition in any given graph.

20 The proof is very similar to that of Theorem 2.1. Let M be a maximum symmetric matching of G. For each δ i, let U i be the set of vertices of degree i. If U i is odd, then U i contains a vertex, say x i, that is not covered by M. We choose a balanced bipartition V 1 and V 2 with the properties that (i) U i V 1 = U i V 2 if U i is even, and U i \ {x i } V 1 = U i \ {x i } V 2 if U i is odd, (ii) for each edge uv M, {u, v} V j for j = 1, 2, 20 of 38 (iii) beginning from the one with the smallest degree, the selected vertices x s alternatively belong to V 1 and V 2, and (iv) subject to (i)-(iii), e(v 1, V 2 ) is maximum. We omit the detailed calculations.

21 Below is a corollary of Theorem 3.1 that says conjecture 1 holds for graphs of which m is not very small. Corollary 3.2 (Xu, Yan and Yu, DM 2010 inpress) Let G be a graph with m edges. If m 3n or δ(g) 5, then G admits a balanced bipartition V 1, V 2 such that max{e(v 1 ), e(v 2 )} m/3. Proof. Let n denote the number of vertices in G. If m 3n then (G) n 1 < m/3; and it follows from Theroem 3.1 that G admits a balanced bipartition V 1, V 2 such that max{e(v 1 ), e(v 2 )} (m + )/4 < m/3. 21 of 38

22 Now assume m 3n 1. Then δ(g) = 5. Let t denote the number of vertices of degree 5. We may assume t n 1; for otherwise G is a regular graph and the assertion follows from the result of Bollobás and Scott [5]. By Theorem 3.1, G admits a balanced bipartition V 1, V 2 such that max{e(v 1 ), e(v 2 )} (m + )/4. If (m + )/4 m/3, we are done. So we may assume that (m + )/4 > m/3, i.e., > m/3. Then 2m + 6(n t 1) + 5t > m/3 + 6n t 6 m/3 + 5n 5 (since t n 1). So m/3 > n 1, and hence > m/3 > n 1, a contradiction. 22 of 38

23 Next consequence of Theorem 3.1 answers a problem of Bollobás and Scott in [3]. Corollary 3.3 Let G be a graph with m edges and n vertices. Suppose (G) = o(n), or δ(g) as n. Then G admits a balanced bipartition V 1, V 2 such that max{e(v 1 ), e(v 2 )} (1 + o(1))m/4. There exist examples showing that the conditions in Corollary 3.3 is best possible in the sense that it does not hold for graphs G with (G) = Θ(n) and δ(g) = O(1). 23 of 38

24 4 Balanced bipartition of plane graphs Now, we consider the balanced bipartition problems on plane graphs. Recall that the folklore conjecture claim that every plane graph of order n admits a balanced bipartition V 1, V 2 with e(v 1, V 2 ) n. Our Theorem 2.2 asserts that every graph G of order n and size m admits a balanced bipartition V 1, V 2 such that e(v 1, V 2 ) 1 2 (m + n 2 M ), 24 of 38 where M is a maximum matching of G c.

25 Let G be a graph of order n and clique number c. Since for each maximal matching M of G c, the set of vertices uncovered by M is an independent set in G c, and hence has size at most c. Therefore, G c has a matching of size at least n c 2. Since n 2 n c 2 = c 2, as a direct consequence of Theorem 2.2, we have 25 of 38 Corollary 4.1 Let G be a graph with m edges. Then, G admits a balanced bipartition V 1, V 2 such that e(v 1, V 2 ) 1 2 (m + c(g) 2 ), where c(g) is the clique number of G.

26 A simple calculation by Euler s formula on the sphere shows that every connected triangle-free planar graph of order n 3 has at most 2n 4 edges. As a consequence of Corollary 4.1, every connected triangle-free planar graph of order n 3 has a balanced bipartite subgraph with at most n 2 edges. This bound is sharp as evidenced by the cycle of length of 38 Theorem 4.2 Every connected triangle-free planar graph of order n 3 has a balanced cut with at most n 2 edges. The extremal graphs are exactly K 2,l, where l 2 is an integer.

27 On the plane graphs without separating triangle, the following theorem gives an upper bound that nearly confirms Conjecture 2 on such graphs. 27 of 38 Theorem 4.3 Let G be a planar graph of order n. If G contains no separating triangles, then G admits a balanced bipartition V 1, V 2 such that e(v 1, V 2 ) n + 1.

28 The proof of Theorem 4.3 depends on the following two technical lemmas. Lemma 4.4 Let G be a connected planar graph without separating triangle. Then G is a wheel of order at least 5, or G is obtained from a wheel of order at least 5 by deleting an edge from its ring, or G is a spanning subgraph of a maximal planar graph without separating triangle. Lemma 4.5 Let G be a planar graph with n vertices. If G is Hamiltonian, then G admits a balanced bipartition V 1 and V 2 such that e(v 1, V 2 ) n of 38

29 Let G be a planar graph of order n and containing no separating triangles. If G is a wheel, or a graph obtained from a wheel by removing an edge from its ring, then one can easily find a balanced bipartition of cut size at most n in G. Otherwise, there exists a maximal planar graph H such that H contains no separating triangles and V (H) = V (G) by Lemma 4.4. Therefore, H is Hamiltonian by Whitney s Theorem [13], and hence has a balanced bipartition of cut size at most n + 1 by Lemma of 38

30 5 Some open problems Theorems 2.1 and 2.2 give a lower bound and an upper bound on the maximum balanced bipartitions and minimum balanced bipartitions of graphs, respectively. A natural question is the following formulation of the Bollobás-Scott question mentioned above. Problem 3 (Xu, Yan and Yu, DM 2010 inpress) Are there functions f(m) and g(m) such that for every graph G of size m, max{e(v 1, V 2 ) V 1, V 2 is a balanced bipartition of G} m 2 + f(m), and min{e(v 1, V 2 ) V 1, V 2 is a balanced bipartition of G} m 2 + g(m)? 30 of 38

31 In [14], it is proved that every graph G with (G) 7 5 δ(g) admits a balance bipartition V 1, V 2 such that max{e(v 1 ), e(v 2 )} E(G) /3. Its proof actually shows that for any graph G with (G) 7 5 δ(g), any balanced bipartition V 1, V 2 of V (G) with e(v 1, V 2 ) maximum (among all balanced bipartitions) must satisfy e(v i ) E(G) /3. Theorem 2.1 ensures that the balanced bipartition V 1, V 2 constructed in [14] has e(v 1, V 2 ) ( E(G) + m(g))/2, where m(g) denotes the number of edges in a maximum matching of G. 31 of 38

32 An example given in [14] shows that there exist graphs G such that for any maximum balanced bipartition V 1, V 2 of V (G), max{e(v 1 ), e(v 2 )} > E(G) /3. So it would be interesting to know which graphs G admits a maximum balanced bipartition V 1, V 2 such that 32 of 38 max{e(v 1 ), e(v 2 )} E(G) /3?

33 To be precisely, we may ask the following problem. Problem 4 (Xu, Yan and Yu [14]) What is the largest constant c such that for any graph G with (G) cδ(g), if V 1, V 2 is a balanced bipartition of V (G) with e(v 1, V 2) maximum then max{e(v 1 ), e(v 2 )} e(g) /3? The example in [14] shows that c < 13/4. 33 of 38

34 Conjecture 1 is still open in the case that 2 δ(g) 4. So, one problem on this direction is to prove or disprove this conjecture. If Conjecture 1 is true, then δ(g) 2 guarantees the existence of a balanced bipartition V 1, V 2 with max{e(v 1 ), e(v 2 )} m/3, and the constant 1/3 is best possible as evidenced by K 3. Can we decrease the constant 1/3 by increasing minimum degree? The regular graphs of even order tell us that the constant is at least 1/4. Bollobás and Scott proposed the following problems [3]. Problem 5 (Bollobás and Scott [3]) What is the smallest constant c(k) such that every graph with minimum degree at least k has a balanced bipartition with at most c(k)e(g) edges in each class? 34 of 38 This problem is very hard!

35 Below is another conjecture of Bollobás and Scott on balanced bipartition problems. Conjecture 6 (Bollobás and Scott [3]) Every graph has a balanced bipartition V 1, V 2 with N(v) V 1 N(v) V for every v V 1 and 35 of 38 N(v) V 2 N(v) V for every v V 2. This is also a very hard problem!

36 On the minimum balanced bipartitions of plane graphs, Conjecture 2 is still open. Maybe it is not a bad result if one can find a constant p such that every plane graph of order n has a balanced bipartition V 1, V 2 with e(v 1, V 2 ) n + p. 36 of 38

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