Biholes in balanced bipartite graphs
aa r X i v : . [ m a t h . C O ] A p r Biholes in balanced bipartite graphs
Stefan Ehard, Elena Mohr, and Dieter Rautenbach
Institute of Optimization and Operations Research, Ulm University, Germany { stefan.ehard,elena.mohr,dieter.rautenbach } @uni-ulm.de Abstract
A bihole in a bipartite graph G with partite sets A and B is an independent set I in G with | I ∩ A | = | I ∩ B | . We prove lower bounds on the largest order of biholes in balanced bipartitegraphs subject to conditions involving the vertex degrees and the average degree. Keywords: bihole; independent set
MSC 2020 classification:
In [1,2] Axenovich et al. study biholes defined as independent sets in bipartite graphs containing equallymany vertices from both parts of a fixed bipartition. They present several lower bounds on the orderof largest biholes subject to degree conditions. Here we pursue some of the questions motivated by [2].For a detailed discussion of the motivation of biholes, we refer to [2]. First, we collect some notationand definitions. We consider only finite, simple, and undirected graphs. For a graph G , we denote thevertex set, the edge set, the order, and the size by V ( G ), E ( G ), n ( G ), and m ( G ), respectively. Let G be a bipartite graph with partite sets A and B . A bihole of order k in G is an independent set I in G with | I ∩ A | = | I ∩ B | = k . Note that the definition of a bihole tacitly requires to fix a bipartitionof G , which is unique only if G is connected. Note furthermore that the order of a bihole I is halfthe cardinality of the set I . Let ˜ α ( G ) be the largest order of a bihole in G . A bipartite graph G withpartite sets A and B is balanced if | A | = | B | . For an integer k , let [ k ] be the set of positive integers atmost k , and let [ k ] = { } ∪ [ k ].For positive integers n and ∆, Axenovich et al. [2] define f ( n, ∆) as the largest integer k such thatevery bipartite graph G with partite sets A and B satisfying • | A | = | B | = n , and • the degree d G ( u ) of every vertex u from A is at most ∆,has a bihole of order k . Similarly, they define f ∗ ( n, ∆) as the largest integer k such that every balancedbipartite graph G of order 2 n and maximum degree at most ∆, has a bihole of order k . The definitionsimmediately imply f ( n, ∆) ≤ f ∗ ( n, ∆).In [2] Axenovich et al. show the following results for integers n and ∆ with n ≥ ∆ ≥ f ( n,
2) = l n m − , (1) f ( n, ∆) ≥ (cid:22) n − (cid:23) , (2)1 ( n, ∆) = Θ (cid:18) ln ∆∆ n (cid:19) for large but fixed ∆ and n sufficiently large, and (3)0 . n < f ( n, ≤ f ∗ ( n, < . n for n sufficiently large. (4)They explicitly ask for the value of f ( n,
3) for sufficiently large n .While the parameters f ( n, ∆) and f ∗ ( n, ∆) might appear closely related to the independencenumber α ( G ) of a graph G , and one might be tempted to expect a similar behavior, the requirement tocontain equally many vertices from both partite sets imposes a strict condition. In fact, balancing theintersections with the partite sets seems to be one of the challenges in proofs about these parameters.The following three tight lower bounds on the independence number α ( G ) of a graph G withaverage degree d = m ( G ) n ( G ) and maximum degree at most ∆ are well known [3, 5, 6]: α ( G ) ≥ X u ∈ V ( G ) d G ( u ) + 1 ≥ nd + 1 ≥ n ∆ + 1 . (5)The inequality (2) translates the final bound in (5) from independent sets to biholes, but (3) indicatesthat asymptotically stronger lower bounds hold. The result (3) implies the following similar resultinvolving the average degree. Proposition 1.
There exists a real d such that, for every real d ≥ d , there is some integer n ( d ) such that, for every integer n ≥ n ( d ) , the following statement holds: If G is a balanced bipartite graphof order n that has at most dn edges, then ˜ α ( G ) ≥ ln( d )8 d n .Proof. Axenovich et al. [2] show the following:
There exists an integer ∆ such that, for every integer ∆ ≥ ∆ , there is some integer n (∆) such that f ( n, ∆) ≥ ln(∆) n for every n ≥ n (∆) . Let d = max (cid:8) , ∆ +12 (cid:9) and, for every real d ≥ d , let n ( d ) = 2 n ( ⌊ d ⌋ ). Now, let d be any realat least d , and let n be any integer at least n ( d ). Let G be a balanced bipartite graph of order2 n that has at most dn edges. Let A and B be the partite sets of G . Let n > d be the numberof vertices u in A with d G ( u ) > d . Since 2 dn > d ≤ dn , we have n > d ≤ n , which implies that A ′ = { u ∈ A : d G ( u ) ≤ ⌊ d ⌋} contains at least n vertices. Let G ′ arise from G by removing all verticesin A \ A ′ from A as well as any | A \ A ′ | vertices from B . Clearly, the graph G ′ is a balanced bipartitegraph of order 2 n ′ with n ′ ≥ n such that d G ′ ( u ) ≤ ⌊ d ⌋ for every vertex in A ′ . Since d ≥ ⌊ d ⌋ ≥ ∆ ,and n ′ ≥ n ( d )2 = n ( ⌊ d ⌋ ), the result from [2] implies˜ α ( G ) ≥ ˜ α ( G ′ ) ≥ f ( n ′ , ⌊ d ⌋ ) ≥ ln( ⌊ d ⌋ ) n ′ ⌊ d ⌋ ≥ ln( d ) n d . Inspired by the second bound in (5), we prove the following, which, in view of Proposition 1, isinteresting for small values of d or n . Theorem 2. If G is a balanced bipartite graph of order n that has at most dn edges for somenon-negative real d , then ˜ α ( G ) ≥ nd + 1 − . (6)2urthermore, we contribute a small improvement of the lower bound on f ( n,
3) from [2]. Therefore,we need the following refined version of f ( n, ∆): For non-negative integers d < d < . . . < d ℓ and n , n , . . . , n ℓ , let ˜ α ( d n , d n , . . . , d n ℓ ℓ ) be the largest k such that every bipartite graph G with partitesets A and B such that • | A | = | B | = n + n + · · · + n ℓ , and • n i = |{ u ∈ A : d G ( u ) = d i }| for every i ∈ [ ℓ ],has a bihole of order k . For the considered graphs, the sequence d . . . d | {z } n . . . d ℓ . . . d ℓ | {z } n ℓ is the degreesequence of the vertices in A . Note that f ( n, ∆) = min n ˜ α (0 n , . . . , ∆ n ∆ ) : n , . . . , n ∆ ∈ N with n = n + · · · + n ∆ o . Our next result can be considered to be a refinement of (1).
Theorem 3. ˜ α (0 n , n , n ) ≥ n + ( n + n ) − . Finally, combining Theorem 3 with the approach of Axenovich et al. [2], allows to slightly improvetheir lower bound on f ( n,
3) as follows.
Theorem 4.
For every ǫ ≥ , there is some n such that f ( n, ≥ (cid:0) . − ǫ (cid:1) n for every n ≥ n . The proofs of the stated results as well as of further auxiliary statements are given in the followingsection.
We begin with a restricted analogue of Theorem 3.
Lemma 5. ˜ α (0 n , n ) ≥ n + n − .Proof. Let G be a bipartite graph with partite sets A and B such that • | A | = | B | = n + n , and • n i = |{ u ∈ A : d G ( u ) = i }| for i ∈ { , } .Let G , . . . , G k be the components of G that are of order more than 2. Each G i is a star with n ,i ≥ A and a center vertex from B . It follows that G contains ℓ = n − k P i =1 n ,i componentsthat are K ’s, and B contains n + k P i =1 ( n ,i −
1) isolated vertices. Now, there is a bihole I in G containing • all n isolated vertices from A , • at least ℓ − vertices of degree 1 from A as well as at least ℓ − vertices of degree 1 from B ; allcoming from K components, • k P i =1 ( n ,i −
1) vertices of degree 1 from A ; coming from the G i ’s, and • all n + k P i =1 ( n ,i −
1) isolated vertices from B .3ince k ≤ n − ℓ , we obtain that I has order at least n + ℓ −
12 + k X i =1 ( n ,i −
1) = n + ℓ −
12 + ( n − ℓ − k ) ≥ n + n − ℓ − n − ℓ − n + n − , which completes the proof.Now, we proceed to the proof of Theorem 2. Proof of Theorem 2.
Suppose, for a contradiction, that G is a counterexample of minimum order2 n . Let ∆ A be the maximum degree of the vertices in A . First, we assume that ∆ A <
2. Let A contain n i vertices of degree i for i ∈ { , } . Since G has n edges, we have d ≥ n n + n . Now, since n + n ≥ n + n n n n +1 , Lemma 5 implies˜ α ( G ) ≥ n + n − ≥ n + n n n + n + 1 −
12 = nd + 1 − , and (6) follows. Next, we assume that 2 ≤ ∆ A < d + 1. In this case, the inequality (2) implies˜ α ( G ) ≥ (cid:22) n − A (cid:23) > n ∆ A − > nd + 1 − . Finally, we may assume that ∆ A ≥ d + 1. By symmetry, we may also assume that the maximumdegree ∆ B of the vertices in B satisfies ∆ B ≥ d + 1. Let u ∈ A and v ∈ B be vertices of degree atleast d + 1. The graph G ′ = G − { u, v } is balanced with partite sets of order n −
1, and at most nd − d G ( u ) − d G ( v ) + 1 ≤ nd − d − G , the graph G ′ is no counterexample,and we obtain ˜ α ( G ) ≥ ˜ α ( G ′ ) ≥ n − nd − d − n − + 1 − n − ( d + 1)( n − − ≥ nd + 1 − , where we use ( n − ≥ n ( n − d = 2, and gives a better additive constant. Proposition 6. If G is a balanced bipartite graph of order n ≥ that has at most n edges, then ˜ α ( G ) ≥ n − .Proof. We prove the statement by induction on n . Let A and B be the partite sets of G . For n = 2, thestatement is trivial. Now, let n ≥
3. Let δ A = min { d G ( u ) : u ∈ A } , ∆ A = max { d G ( u ) : u ∈ A } , anddefine δ B as well as ∆ B analogously. By the result (1) of Axenovich et al. [2], and, since n − ≥ n − ,we may assume that ∆ A , ∆ B ≥
3. Since G has at most 2 n edges, this implies δ A , δ B ≤ δ A = 0. Let u be an isolated vertex from A . Let v be a vertex of degree δ B from B . Let u ′ be a vertex from A \{ u } of largest possible degree such that N G ( v ) ⊆ { u ′ } . Let v ′ be a vertexon degree ∆ B from B . Note that m ( G ′ ) ≤ n − G ′ = G − { u, v, u ′ , v ′ } has a bihole I ′ of order at least n − − . Since adding u and v to I ′ yields a bihole in G , the desiredstatement follows. Hence, by symmetry, we may assume that δ A = δ B = 1.4ext, suppose that there are non-adjacent vertices u from A and v from B that are both of degree1. Let v ′ be the neighbor of u , and let u ′ be the neighbor of v . If d G ( u ′ ) ≥
3, then, by induction, thegraph G ′ = G − { u, v, u ′ , v ′ } has a bihole I ′ of order at least n − − . Adding u and v to I ′ yields abihole of G of the desired order. Hence, by symmetry, we may assume that d G ( u ′ ) , d G ( v ′ ) ≤
2. Let u ′′ be a vertex of degree ∆ A from A , and let v ′′ be a vertex of degree ∆ B from B . Let G ′′ be thegraph G − { u, v, u ′ , v ′ , u ′′ , v ′′ } . It is easy to see that m ( G ′′ ) ≤ n − G ′′ has a bihole I ′′ of order at least n − − . Adding u and v to I ′′ yields a bihole of G of the desiredorder. Hence, we may assume that A and B both contain unique vertices of degree 1, say u and v ,respectively, and that u and v are adjacent. Since n ≥ m ( G ) ≤ n , and ∆ A ≥
3, there is a vertex u ′ of degree 2 in A . Let v ′ and v ′′ be the two neighbors of u ′ . Let u ′′ be a vertex of degree ∆ A from A .Let G ′′ be the graph G − { u, v, u ′ , v ′ , u ′′ , v ′′ } . It is easy to see that m ( G ′′ ) ≤ n − G ′′ has a bihole I ′′ of order at least n − − . Adding v and u ′ to I ′′ yields a bihole of G ofthe desired order, which completes the proof.Our next goal is the proof of Theorem 3, which refines (1), and allows to slightly improve the lowerbound on f ( n, Lemma 7. If G is a connected bipartite graph with partite sets A and B such that | A | < | B | and everyvertex in A has degree at most , then G is a tree, | B | = | A | + 1 , and every vertex in A has degreeexactly . Furthermore, for every i in [ | A | ] , there is an independent set I in G with | I ∩ A | = i and | I ∩ B | = | A | − i .Proof. Since every vertex in A has degree at most 2, the graph G has at most 2 | A | edges. Since G isconnected, it has at least | A | + | B | − ≥ | A | edges. It follows that G has exactly 2 | A | edges, everyvertex in A has degree exactly 2, | B | = | A | + 1, and G is a tree.We prove the existence of the desired independent sets by induction on | A | . For | A | = 1, thestatement is trivial. Now, let | A | ≥
2. Clearly, choosing I as A yields | I ∩ A | = | A | and | I ∩ B | = | A | − | A | = 0, that is, the statement is trivial for i = | A | . Now, let i ∈ [ | A | − . Let u be a vertexof degree 1, and let v be its unique neighbor. By induction applied to G ′ = G − { u, v } , the graph G ′ has an independent set I ′ with | I ′ ∩ A | = i and | I ′ ∩ B | = ( | A | − − i , and adding u to I ′ yields thedesired independent set. Proof of Theorem 3.
By induction on n , we show that every bipartite graph G with partite sets A and B such that • | A | = | B | = n + n + n , and • n i = |{ u ∈ A : d G ( u ) = i }| for every i ∈ [2] ,has a bihole of order at least n + ( n + n ) − . If n ≤
3, then G has a bihole of order at least˜ α (0 n , n , n ) ≥ ˜ α (0 , , n + n + n ) ( ) ≥ n + n + n − ≥ n + 12 ( n + n ) − . Now, let n ≥
4. Let u , . . . , u be four isolated vertices from A . Let G , . . . , G r be the components of G with | V ( G i ) ∩ A | < | V ( G i ) ∩ B | . Since | A | = | B | and n ≥
4, there is at least one such component,that is, we have r ≥
1. By Lemma 7, each G i is a tree with | V ( G i ) ∩ B | = | V ( G i ) ∩ A | + 1, whichimplies r ≥ n ≥
4. Let A i = V ( G i ) ∩ A , B i = V ( G i ) ∩ B , and a i = | A i | , for i in [4]. Clearly, wemay assume that a ≤ a ≤ a ≤ a . If B contains an isolated vertex v , then, applying induction to5 ′ = G − { u , v } , we obtain that G ′ has a bihole of order at least ( n −
1) + ( n + n ) − , andadding u and v yields a bihole of more than the desired order. Hence, we may assume that no vertexin B is isolated, in particular, we have a ≥ a and a have different parities modulo 2. By Lemma 7, there is anindependent set I of G with | I ∩ A | = a + a −
12 and | I ∩ B | = a − a + 12 . By induction, the graph G ′ = G − (cid:0) { u , u } ∪ V ( G ) ∪ V ( G ) (cid:1) has a bihole I ′ of order at least ( n −
2) + ( n + n − a − a ) − . Now, the set (cid:0) { u , u } ∪ B ∪ I (cid:1) ∪ I ′ is a bihole in G of orderat least12 (cid:0) a + 1) + a (cid:1) + (cid:18)
34 ( n −
2) + 12 ( n + n − a − a ) − (cid:19) = 34 n + 12 ( n + n ) − . Hence, we may assume that a and a have the same parity modulo 2, and, by symmetry, that also a and a have the same parity modulo 2. Note that a + a − ∈ [ | A | ] . By Lemma 7, there is anindependent set I of G with | I ∩ A | = a + a | I ∩ B | = a − a , as well as an independent set I of G with | I ∩ A | = a + a −
22 and | I ∩ B | = a − a + 22 . By induction, the graph G ′′ = G − (cid:0) { u , u , u , u } ∪ V ( G ) ∪ V ( G ) ∪ V ( G ) ∪ V ( G ) (cid:1) has a bihole I ′′ of order at least ( n −
4) + ( n + n − a − a − a − a ) − . Now, the set (cid:0) { u , u , u , u } ∪ B ∪ I ∪ B ∪ I (cid:1) ∪ I ′′ is a bihole in G of order at least12 (cid:0) a + 1) + a + ( a + 1) + a (cid:1) + (cid:18)
34 ( n −
4) + 12 ( n + n − a − a − a − a ) − (cid:19) = 34 n + 12 ( n + n ) − , which completes the proof.The following example shows that the coefficient for n in Theorem 3 is best possible: For an eveninteger i , let the bipartite graph G have partite sets A and B and exactly 2 i components such thatthere are i isolated vertices that all belong to A , and i paths P , . . . , P i , each of order 4 i + 1, whoseendpoints all belong to B . Note that | A | = | B | = i + 2 i , n = i , and n = 2 i . Let I be a largestbihole in G . From every path P i , at most 2 i + 1 vertices can belong to I , and, if 2 i + 1 vertices belongto I , then V ( P i ) ∩ I ⊆ B . If in more than i of the paths P i , at least 2 i + 1 vertices belong to I , then | I ∩ A | ≤ (cid:18) i − (cid:19) i + i = i − i < i + 52 i + 1 = (cid:18) i (cid:19) (2 i + 1) ≤ | I ∩ B | , which is a contradiction. Hence, in at most i of the paths P i , at least 2 i + 1 vertices belong to I ,which implies | I | ≤ (cid:18) i + (2 i + 1) i i i (cid:19) = i + 34 i = 34 n + 12 n .
6t seems a challenging problem to determine the value ˜ α (0 n , n , n ) exactly for all choices of n , n ,and n . In fact, depending on the relative values of the n i , they should contribute to this value withdifferent coefficients. If, for instance, n = 0, then, by Lemma 5, the coefficient of n is 1 rather than as in Theorem 3.For the next proof, we need the following Simple Concentration Bound [4]:
Let X be a random variable determined by n independent trials T , . . . , T n such that chang-ing the outcome of any one trial can affect X by at most c , then P h | X − E [ X ] | > t i ≤ e − t c n for every t > . (7) Proof of Theorem 4.
Let G be a bipartite graph with partite sets A and B such that | A | = | B | = n and every vertex in A has degree at most 3. We need to show that G has a bihole of order at least (cid:0) . − o ( n ) (cid:1) n . Therefore, let ǫ be such that 0 < ǫ <
12 ln(8) < .
25. Let B large be the set of verticesin B of degree more than ǫ / √ n , and let B small = B \ B large . Since G has at most 3 n edges, we have | B large | ≤ √ nǫ / . Let G (1) arise from G by removing B large as well as any set of | B large | vertices from A .Let n (1) i = (cid:12)(cid:12)(cid:12)n u ∈ V ( G (1) ) ∩ A : d G (1) ( u ) = i o(cid:12)(cid:12)(cid:12) for i ∈ [3] , and let n (1) = (cid:12)(cid:12) V ( G (1) ) ∩ A (cid:12)(cid:12) , that is, n (1) = n (1)0 + n (1)1 + n (1)2 + n (1)3 = n − | B large | ≥ (cid:18) − ǫ / √ n (cid:19) n. Let p be the real solution of the equation p = (1 − p ) , that is, p ≈ . B (1) be a randomsubset of B small that arises by adding each of the n (1) vertices in B small to the set B (1) independentlyat random with probability p . Let G (2) arise from G (1) by removing B (1) , let b (1) = | B (1) | , and let n (2) i = (cid:12)(cid:12)(cid:12)n u ∈ V ( G (2) ) ∩ A : d G (2) ( u ) = i o(cid:12)(cid:12)(cid:12) for i ∈ [3] .For the random variables b (1) , n (2)0 , and n (2)3 , we obtain E h b (1) i = pn (1) , E h n (2)0 i = n (1)0 + pn (1)1 + p n (1)2 + p n (1)3 ≥ p n (1) , E h n (2)3 i = (1 − p ) n (1)3 ≤ (1 − p ) n (1) . Applying (7) with c = ǫ / √ n in each case, and using ǫ <
12 ln(8) , we obtain P h (cid:12)(cid:12)(cid:12) b (1) − E h b (1) i(cid:12)(cid:12)(cid:12) > ǫn (1) i ≤ e − ( ǫn (1) ) ǫ nn (1) ≤ e − − ǫ / √ n ! ǫ < , P h (cid:12)(cid:12)(cid:12) n (2)0 − E h n (2)0 i(cid:12)(cid:12)(cid:12) > ǫn (1) i < , and P h (cid:12)(cid:12)(cid:12) n (2)3 − E h n (2)3 i(cid:12)(cid:12)(cid:12) > ǫn (1) i < , for n sufficiently large. 7or n sufficiently large, the union bound implies the existence of a choice of B (1) such that b (1) ≤ ( p + ǫ ) n (1) ,n (2)0 ≥ ( p − ǫ ) n (1) , and n (2)3 ≤ (cid:0) (1 − p ) + ǫ (cid:1) n (1) = ( p + ǫ ) n (1) . Let G (3) arise from G (2) by removing • a set containing max n b (1) , n (2)3 o vertices from V ( G (2) ) ∩ A including all vertices from V ( G (2) ) ∩ A that are of degree 3 in G (2) and as few isolated vertices of G (2) as possible, and • a set containing max n b (1) , n (2)3 o vertices from V ( G (2) ) ∩ B .By construction all vertices in V ( G (3) ) ∩ A have degree at most 2 in G (3) . Since( p − ǫ ) n (1) + max n b (1) , n (2)3 o ≤ ( p − ǫ ) n (1) + ( p + ǫ ) n (1) ≤ . n (1) ≤ n (1) , the number n (3)0 of vertices in V ( G (2) ) ∩ A that are isolated in G (3) is at least ( p − ǫ ) n (1) . By Theorem3, the graph G (3) , and, hence, also G , contains a bihole of order at least34 n (3)0 + 12 (cid:16) n (1) − n (3)0 − max n b (1) , n (2)3 o (cid:17) − C ≥
34 ( p − ǫ ) n (1) + 12 (cid:16) n (1) − ( p − ǫ ) n (1) − ( p + ǫ ) n (1) (cid:17) − C ≥ (cid:18)