On splitting digraphs

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1 On splitting digraphs arxiv: v [math.co] 0 Apr 08 Donglei Yang a,, Yandong Bai b,, Guanghui Wang a,, Jianliang Wu a, a School of Mathematics, Shandong University, Jinan, 5000, P. R. China b Department of Applied Mathematics, Northwestern Polytechnical University, Xi an, 7007, P. R. China Abstract In 995, Stiebitz asked the following question: For any positive integers s,t, is there a finite integer f(s,t) such that every digraph D with minimum out-degree at least f(s,t) admits a bipartition (A, B) such that A induces a subdigraph with minimum out-degree at least s and B induces a subdigraph with minimum out-degree at least t? We give an affirmative answer for tournaments, multipartite tournaments, and digraphs with bounded maximum in-degrees. In particular, we show that for every ǫ with 0 < ǫ < /, there exists an integer δ 0 such that every tournament with minimum out-degree at least δ 0 admits a bisection (A,B), so that each vertex has at least(/ ǫ) of its out-neighbors in A, and in B as well. Keywords: Bipartitions of digraphs; Tournaments; Weighted Lovász Local Lemma; Introduction Partitioning an undirected graph (a digraph) into two parts under certain constraints (e.g., see [7] for connectivity constraint, see [5] for chromatic number constraint) has been widely studied due to its important applications in induction arguments. Among them, partitions under degree constraints have attracted special attention and a number of classical results in undirected graphs have been achieved. Lovász [] proved in 966 that every undirected graph with maximum degree s t can be partitioned into two parts such that they induce two subgraphs with maximum degree at most s and at most t, respectively. Stiebitz address: dlyang0@63.com. address: bai@nwpu.edu.cn. Corresponding author. address: ghwang@sdu.edu.cn address: jlwu@sdu.edu.cn.

2 [4] showed in 996 that every undirected graph with minimum degree s t can be partitioned into two parts such that they induce two subgraphs with minimum degree at least s and at least t, respectively. A natural question is whether or not the corresponding assertions for digraphs hold, where the degree is replaced by out-degree. For Lovász s result under maximum degree constraint, Alon [] pointed out that its corresponding assertion fails for digraphs in the following strong sense: For every k, there is a digraph without even cycles, in which all out-degrees are exactly k. It is trivial to prove that for every bipartition of such a digraph, the maximum out-degree in one of the two parts is k. This example was given by Thomassen [7] in 985. How about the corresponding assertion for digraphs with respect to Stiebitz s result? In fact, in 995, Stiebitz [3] proposed the following problem. Problem. For any positive integers s,t, is there a finite integer f(s,t) such that every digraph with minimum out-degree at leastf(s,t) admits a bipartition (A,B), so that A induces a subdigraph with minimum out-degree at least s and B induces a subdigraph with minimum out-degree at least t? This problem was also mentioned in []. For general digraphs, the only known value is f(, ) = 3 from a result of Thomassen [6]. Lichiardopol [0] proved that every tournament with minimum out-degree at least t s 3s admits a bipartition (A,B) such that A induces a subdigraph with minimum out-degree at least s and B induces a subdigraph with minimum out-degree at least t. Kézdy [6] constructed an example showing that f(, ) > 5. For more results about splitting digraphs, the readers are referred to [4, 9]. Particularly, we ask for a bipartition (A,B) with A and B of fixed sizes. A bisection is a bipartition (A,B) with A B. Bollobás and Scott [3] conjectured that every graph G has a bisection (A,B) with d H (v) d G(v) for each v V(G), where H is the subgraph induced by the set of edges between A and B. However, Ji et al. [5] gave an infinite family of counterexamples to this conjecture, which indicates that d G(v) is probably the correct lower bound. This conjecture is widely open and readers are referred to [3, 5, 8]. For digraphs, unfortunately, the same example given by Thomassen [7] indicates that we cannot obtain a bipartition (A,B) such that each vertex in one part has at least one outneighbor in the other part. So we begin to consider whether or not there exists a bisection of any digraph such that the two subdigraphs induced by the two parts have high minimum out-degree, and we propose the following problem. Problem. For any positive integers s,t, is there a finite integer f(s,t) such that every digraph with minimum out-degree at least f(s,t) admits a bisection (A,B), so that A induces

3 a subdigraph with minimum out-degree at least s and B induces a subdigraph with minimum out-degree at least t? In this paper, we give affirmative answers to Problems and for some classes of digraphs. Given a digraph D and a bipartition (A,B) of V(D), let (D) be the maximum in-degree of D and e(a,b) be the number of arcs from A to B. Let D[A] and D[B] be the induced subdigraphs of D on A and B, respectively. For a vertex v V(D), we write d A (v) for the number of out-neighbors of v in A, and d B (v) for the number of out-neighbors of v in B. All digraphs considered here are simple (without loops or multiple arcs). An n-partite tournament with n, or multipartite tournament, is an orientation of a complete n-partite graph, and particularly, a tournament is an orientation of a complete graph. A digraph is strong if, for every two vertices x and y, there exists an (x,y)-path. As for tournaments, we have the desired result in the following strong sense. Theorem. For everyǫwith 0 < ǫ <, there exists an integerδ 0 such that every tournament T with δ (T) δ 0 admits a bisection (A,B) with min{d A (v),d B (v)} ( ǫ)d T (v) for every v V(T). This result gives affirmative answers to Problems and for tournaments. A digraph with minimum out-degree s is s-minimal if any proper subdigraph has minimum out-degree at most s. It is not hard to prove that any s-minimal tournament T with s > 0 is strong. So we have the following result. Corollary. For every ǫ with 0 < ǫ <, there exists an integer δ 0 such that every tournament T with δ (T) δ 0 admits a bipartition (A,B) such that T[A] is strong and d A (v) ( ǫ)d T (v) for every v A, d B (v) ( ǫ)d T (v) for every v B. In fact, we can start with a bisection(a,b) from Theorem, such thatmin{d A (v),d B (v)} ( ǫ)d T (v) for every v V(T). By moving vertices from A to B, we have a minimal subset A A such that d A (v) ( ǫ)d T (v) for every v A. Clearly, T[A ] is strong and d B (v) ( ǫ)d T (v) for every v B = V(T)\A. By the weighted Lovász Local Lemma [], we have the following theorem. Theorem. For every 0 < ǫ <, there exists an integer δ 0 such that every digraph D with δ (D) δ 0 and (D) eǫ (δ (D) ) 8δ (D) ( ǫ)d D (v) for every v V(D). admits a bisection (A,B) with min{d A (v),d B (v)} For bipartite tournaments, we also have an affirmative answer to Problem. 3

4 Theorem 3. For any positive integers s t, if D is a bipartite tournament with δ (D) t (s)4 4s s, then D has a bipartition (A,B) with δ (D[A]) s and δ (D[B]) t. For k-partite tournaments with k >, we derive the following result from similar arguments in the proof of Theorem 3, and here we only give a trivial bound. Corollary. For any positive integers s t and k >, if D is a k-partite tournament with δ (D) tmax{s(s),ks(s)}, then D has a bipartition (A,B) with δ (D[A]) s and δ (D[B]) t. Proofs of Theorem, and 3 Proof of Theorem. Let T be such a tournament with n vertices, and assume n is even. We arbitrarily partition the vertices oft into disjoint pairs{v,w },{v,w },...,{v n/,w n/ } (we allow a singleton when n is odd and deal with it in the similar method) and separate each pair independently and uniformly, then we have a bisection (A,B). For a vertex v A (or B) in the pair {v,w}, let X v be the number of out-neighbors of vertex v in A (or B). We say v is bad if either X v < t := ( ǫ)d T (v) or X v > d T (v) t and denote by X the number of bad vertices. For every v V(T), let a v = {i [ n ] : {v i,w i } N (v)} and b v = {i [ n ] : N (v) {v i,w i } = }. Thus we have d T (v) = a v b v and Pr(X v < t) = Pr(X v > d T (v) t) = 0 when a v t. Consider a v < t, we have t a v ( bv )( bv ) w N (v), Pr(X v < t) = t a v ) bv w N (v). Similarly, we have Pr(X v > d T (v) t) = that t av ( bv i t a v ( bv )( ( bv )( bv ) t a v ( bv )( w N (v), ) bv w N (v). If a v < t, then ( b v ) bv ( i)( bv )( ) bv for each i with 0 i t a v, and it follows )( t a bv v ( ) bv )( t a bv ( ). Let f(a,b) = b ) i ( )b, where a < t and 4

5 ab = d T (v). Now we claim that f(a,b) > f(a,b), in fact, ( ) ( b t a ( ) t a b ( ) ) b f(a,b) f(a,b) = 4 i i ( ) b (( ) ( ) ( b b = t a t a ( ) b ( ) (b )!(b ta) = (b ta)!(t a)! > 0, b t a where the last inequality follows from the fact ab > t and a < t. Thus we have t Pr(X v < t) f(0,d T (v)) = ( d T (v) )( ) d T (v) and t Pr(X v > d T (v) t) ( d T (v) )( ) d T (v). )). Suppose a random variable Y has binomial distribution B(N, ), where N = d T (v). By Chernoff s inequality, we know that Pr(Y E(Y) Nσ) < e Nσ for any positive constant σ. Thus we have t ( d T (v) )( ) d T (v) = Pr(Y d T (v) t) = Pr(Y N N t) < e (v) ) ( ) (d T t d T (v) < e ǫ (d T (v) ), where the last inequality follows the fact that t > t > ǫ d T (v) d T (v) whend T (v). ǫ Now we bound E(X). By the linearity of expectation, E(X) = {Pr(X v > d T (v) t)pr(x v < t)} < e ǫ (d T (v) ). v V (T) v V (T) For every i N, the number of vertices v with i d T (v) < i in T is at most i, and there exists a positive integer i 0 such that e ǫ ( i ) i whenever i i 0. Let δ (T) δ 0 := max{ i 0, ǫ }, we have E(X) < i i 0 i e ǫ ( i ) i i 0 i i. 5

6 Thus there is a bisection of T with no bad vertices, and we are done. From the proof of Theorem, we derive the following corollary. Corollary 3. Every tournament T with δ (T) (o())k admits a bisection (A,B) with min{d A (v),d B (v)} k for every v V(T), where the o()-term tends to zero as k tends to infinity. Proof of Theorem. First we introduce a well-known lemma. Lemma. (The Weighted Local Lemma []) Consider a set B = {A,A,...,A n } of events such that each A i is mutually independent of B (D i {A i }) for some D i B. If we have integers t,t,...,t n and a real number 0 p 4 (a) Pr(A i ) p t i and (b) A j D i (p) t j t i, then with positive probability, none of the events in B occur. such that for each i [n], The proof of Theorem relies on Lemma. We arbitrarily partition the vertices of D into disjoint pairs and separate each pair independently and uniformly, then we have a bisection (V,V ). For a vertex v V(D), let x(v) be the number of out-neighbors of v that are in the same part with v. Let A(v) be the event that either x(v) < s := ( ǫ)d D (v) or x(v) > d D (v) s, and let A = {A(v) : v V(D)} be the set of all bad events. By the same argument as in the proof of Theorem, we have Pr(A(v)) = Pr(x(v) < s)pr(x(v) > d D (v) s) < e ǫ (d D (v) ). Let t v := d D (v) δ (D) be the associated weight. Let p := e ǫ (δ (D) ) and p < 4 whenever δ (D) is sufficiently large. In fact, it suffices to have δ (D) δ 0 := min{δ : e ǫ (δ ) < 4,eǫ (δ ) /8δ δ}. Now it suffices to check that conditions (a) and (b) hold. The condition (a) holds, since Pr(A(v)) < e ǫ (d D (v) ) e ǫ (δ (D) )d D (v)/δ (D) = p tv. Let D(v) be the set of events that are relevant to the event A(v). Therefore A(v) is mutually independent of A (D(v) {A(v)}). We observe that A(v) and A(u) are related only if u, v have common out-neighbors or have neighbors in the same pair. From the observation, we have D(v) d D (v) (D). Since (D) eǫ (δ (D) ) 8δ (D) we have A(w) D(v) (p) tw A(w) D(v) p e ǫ (δ (D) ) D(v) t v. 6 and t v for every v V(D),

7 Now condition (b) holds, and by Lemma, with positive probability, no bad events in A occur. That is, we have a bisection with s x(v) d D (v) s for every v V(D). This observation completes the proof. Proofs of Theorem 3 and Corollary. Recall that a digraph with minimum out-degree s is s-minimal if any proper subdigraph has minimum out-degree at most s. We have the following rough characterization of minimal bipartite and k-partite tournaments (k > ). Lemma. () Every s-minimal bipartite tournament D satisfies V(D) (s)4 4s. () Every s-minimal k-partite tournament D satisfies V(D) < max{s(s),ks(s)} Proof. Let D = (U, W) be an s-minimal bipartite tournament on n vertices. It follows that for any vertex v V(D), there is an arc uv with d D (u) = s. Define a = {v U : d D (v) = s} and b = {v W : d D (v) = s} and without loss of generality let a b. By the fact above, we have s(ab) ab n (ab) 0. So b s. Thus we have n max{g(a,b) = (s )(a b) ab : b s,a sb}. By monotonicity analysis, the optimal solution (x,y) satisfies x = sy, and it follows that n max{g(sy,y) = (s) y sy : y s} (s)4 4s. For an s-minimal k-partite tournament D = (U,U,...,U k ), we denote a i = {v U i : d D (v) = s} for each i {,,...,k} and assume that a a... a k. Similarly, we have s k a i i= i<j k a i a j n k a i. If a k a i, then s k k a i a a i > 0 by the above inequality. Thus we have a < s and n < ks(s). If a > k k a i, then sa a a i > 0 and it follows that k a i < s. By the fact that for any vertex v V(D), there is an arc uv with d D (u) = s, we have a < s k and n < s(s). So n < max{s(s),ks(s)}. Lemma implies Theorem 3 and Corollary directly. 3 Remark i= a i We want to mention that Alon et al. [] obtained a similar result regarding Theorem, and their work was available on arxiv just before we submit our manuscript. The results in two 7

8 papers are finished independently. 4 Acknowledgements We are very grateful to the reviewers for their useful comments. This work was supported by NSFC (Nos , 6304, 4793), the Foundation for Distinguished Young Scholars of Shandong Province (JQ050), and China Postdoctoral Science Foundation (No. 06M590969). References [] N. Alon, Splitting digraphs, Combinatorics, Probability and Computing, 5 (006) [] N. Alon, J. Bang-Jensen, S. Bessy, Out-colorings of digraphs, arxiv: v. [3] B. Bollobás, A. D. Scott, Problems and results on judicious partitions, Random Structures and Algorithms, (00) [4] J. Hou, S. Wu, G. Yan, On bisections of directed graphs, European Journal of Combinatorics, 63 (07) [5] Y. Ji, J. Ma, J. Yan, X. Yu, On problems about judicious bipartitions of graphs, arxiv: [6] A. Kézdy, f(, ) > 5, Personal communication. [7] D. Kühn, D. Osthus, Partitions of graphs with high minimum degree or connectivity, Journal of Combinatorial Theory, Series B, 88 (003) [8] C. Lee, P. Loh, B. Sudakov, Bisections of graphs, Journal of Combinatorial Theory, Series B, 03 (03) [9] C. Lee, P. Loh, B. Sudakov, Judicious partitions of directed graphs, Random Structures and Algorithms, 48 (06) [0] N. Lichiardopol, Vertex-disjoint subtournaments of prescribed minimum outdegree or minimum semidegree: Proof for tournaments of a conjecture of Stiebitz, International Journal of Combinatorics, (0). 8

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