aa r X i v : . [ m a t h . C O ] M a y On the bipartite graph packing problem
B´alint V´as´arhelyi ∗ August 13, 2018
Abstract
The graph packing problem is a well-known area in graph theory.We consider a bipartite version and give almost tight conditions onthe packability of two bipartite sequences.
Keywords: graph packing, bipartite, degree sequence
We consider only simple graphs. Throughout the paper we use commongraph theory notations: d G ( v ) (or briefly, if G is understood from thecontext, d ( v )) is the degree of v in G , and ∆( G ) is the maximal and δ ( G )is the minimal degree of G , and e ( X, Y ) is the number of edges between X and Y for X ∩ Y = ∅ . For any function f on V let f ( X ) = P v ∈ X f ( v ) forevery X ⊆ V . π ( G ) is the degree sequence of G . Let G and H be two graphs on n vertices. We say that G and H pack ifand only if K n contains edge-disjoint copies of G and H as subgraphs.The graph packing problem can be formulated as an embedding prob-lem, too. G and H pack if and only if H is a subgraph of G ( H ⊆ G ).A classical result is the theorem of Sauer and Spencer. ∗ Szegedi Tudom´anyegyetem, Bolyai Int´ezet. Aradi v´ertan´uk tere 1., Szeged, 6720,Hungary, [email protected] , Supported by T ´AMOP-4.2.2.B-15/1/KONV-2015-0006 heorem 1 (Sauer, Spencer [19]) . Let G and G be graphs on n verticeswith maximum degrees ∆ and ∆ , respectively. If ∆ ∆ < n , then G and G pack. Many questions in graph theory can be formulated as special packing prob-lems, see [11]. The main topic of the paper is a type of these packing ques-tions, which is called degree sequence packing to be defined in the nextsection. Some results in this field are similar to that of Sauer and Spencer(Theorem 1).The structure of the paper is as it follows. First, we define the degreesequence packing problem, and survey some results. Next, we state andprove our main result and also show that it is tight. In particular, weimprove a bound given by Diemunsch et al. [4] Finally, we consider somecorollaries of our main theorem.
Let π = ( d , . . . , d n ) be a graphic sequence, which means that there is asimple graph G with vertices { v , . . . , v n } such that d ( v i ) = d i . We say that G represents π .Havel [9] and Hakimi [8] gave a characterization of graphic sequences. Theorem 2 (Hakimi [8]) . Let π = { a , . . . , a n } be a sequence of integerssuch that n − ≥ a ≥ · · · ≥ a n ≥ . Then π is graphic if and only if bydeleting any term a i and subtracting 1 from the first a i terms the remaininglist is also graphic. Kleitman and Wang [12] extended this result to directed graphs.Two graphic sequences π and π pack if there are graphs G and G representing π and π , respectively, such that G and G pack. Obviously,the order does not matter.There is an alternative definition to the packability of two graphicsequences. π and π pack with a fixed order if there are graphs G =( V, E ) and H = ( V, E ) with V ( { v , . . . , v n } ) such that d G ( v i ) = π ( i ) and d H ( v i ) = π ( i ) for all i = 1 , . . . , n . 2 detailed study of degree sequence packing we refer to Chapter 3 ofSeacrest’s PhD Thesis [20].One of the first results in (unordered or fixed order) degree sequencepacking is the Lov´asz–Kundu Theorem [15, 14]. Theorem 3 (Kundu [14]) . A graphic sequence π = ( d , . . . , d n ) has arealization containing a k -regular subgraph if and only if π − k = ( d − k, . . . , d n − k ) is graphic. Though we use the first definition, we give a result for the latter. Let∆ i = ∆( π i ) the largest degree and δ i = δ ( π i ) the smallest degree of π i for i = 1 , π + π they mean the vector sum of (theordered) π and π . Theorem 4 (Busch et al. [2]) . Let π and π be graphic sequences of length n with ∆ = ∆( π + π ) and δ = δ ( π + π ) . If ∆ ≤ √ δn − ( δ − , then π and π pack with a fixed oreder. When δ = 1 , strict inequality is required. Diemunsch et al. [4] showed a condition for (unordered) graphic sequences.
Theorem 5 (Diemunsch et al. [4]) . Let π and π be graphic sequences oflength n with ∆ ≥ ∆ and δ ≥ .If (∆ + 1)(∆ + δ ) ≤ δ n + 1 , when ∆ + 2 ≥ ∆ + δ , and (∆ + 1 + ∆ + δ ) ≤ δ n + 1 , when ∆ + 2 < ∆ + δ , (1) then π and π pack. We study the bipartite packing problem as it is formulated by Catlin [3],Hajnal and Szegedy [7] and was used by Hajnal for proving deep results incomplexity theory of decision trees [6].Let G = ( A, B ; E ) and G = ( S, T ; E ) bipartite graphs with | A | = | S | = m and | B | = | T | = n . They pack in the bipartite sense (i.e. they3ave a bipartite packing ) if there are edge-disjoint copies of G and G in K m,n .Let us define the bigraphic sequence packing problem . We say that asequence π = ( a , . . . , a m , b , . . . , b n ) is bigraphic , if π is the degree sequenceof a bipartite graph G with vertex class sizes m and n , respectively [21].Two bigraphic sequences π and π without a fixed order pack , if thereare edge-disjoint bipartite graphs G and G with degree sequences π and π , respectively, such that G and G pack in the bipartite sense.Similarly to general graphic sequences, we can also define the packingwith a fixed order.Diemunsch et al. [4] show the following for bigraphic sequences: Theorem 6 (Diemunsch et al. [4]) . Let π and π be bigraphic sequenceswith classes of size r and s . Let ∆ ≤ ∆ and δ ≥ . If ∆ ∆ ≤ r + s , (2) then π and π pack. The following lemma, formulated by Gale [5] and Ryser [18], will be useful.We present the lemma in the form as discussed in Lov´asz, Exercise 16 ofChapter 7 [16].
Lemma 7 (Lov´asz [16]) . Let G be a bipartite graph and π a bigraphicsequence on ( A, B ) . π ( X ) ≤ e G ( X, Y ) + π ( Y ) ∀ X ⊆ A, ∀ Y ⊆ B, (3) then π can be embedded into G with a fixed order. For more results in this field, we refer the reader to the monography onfactor theory of Yu and Liu [17].
Theorem 8.
For every ε ∈ (0 , ) there is an n = n ( ε ) such that if n > n , and G ( A, B ) and H ( S, T ) are bipartite graphs with | A | = | B | = | S | = | T | = n and the following conditions hold, then H ⊆ G . Condition 1: d G ( x ) > (cid:0) + ε (cid:1) n holds for all x ∈ A ∪ B d H ( x ) < ε n log n holds for all x ∈ S , Condition 3: d H ( y ) = 1 holds for all y ∈ T . We prove Theorem 8 in the next section. First we indicate why we havethe bounds in Conditions 1 and 2.Condition 1 of Theorem 8 is necessary. Suppose that n − < d G ( x ).That allows G = K n +1 , n − ∪ K n − , n +1 . For all ε > n suchthat if n > n degrees are higher than (cid:0) − ε (cid:1) n , but there is no perfectmatching (i.e. 1-factor) in the graph.Condition 2 is necessary as well. To show it, we give an example. Let G = G ( n, n, p ) a random bipartite graph with p > . n . Let H ( S, T ) be the following bipartite graph: each vertex in T hasdegree 1. In S all vertices have degree 0, except log nc vertices with degree cn log n . The graph H cannot be embedded into G , which follows from theexample of Koml´os et al. [13]Before proving Theorem 8 we compare our main theorem with the pre-vious results. Remark 9.
There are graphs which can be packed using Theorem 8, butnot with Theorem 1.Indeed, ∆ > n and we can choose ∆ >
1. Thus, ∆ ∆ > n . However,with Theorem 8 we can pack G and H . Remark 10.
There are graphs which can be packed using Theorem 8, butnot with Theorem 5.Let π = π ( H ) and π = π ( G ). δ = 1 and ∆ ≤ n
100 log n .If ∆ ≈ n , then ∆ + 2 ≥ ∆ + δ .Furthermore, (∆ + 1)(∆ + δ ) ≈ n · nc log n ≫ n. (4)Although the conditions of Theorem 5 are not satisfied, π and π still pack. Remark 11.
There are graphs which can be packed using Theorem 8, butnot with Theorem 6. 5et π = π ( H ) and π = π ( G ), as above. The conditions of Theorem 6are not satisfied, however, Theorem 8 gives a packing of them.As it is transparent, our main theorem can guarantee packings in cases,that were far beyond reach by the previous tecniques. We formulate the key technical result for the proof of Theorem 8 in thefollowing lemma.
Lemma 12.
Let ε and c such that in Theorem 8. Let G and H be bipartitegraphs with classes Z and W of sizes z and n , respectively, where z > ε .Suppose that(i) d G ( x ) > (cid:0) + ε (cid:1) n for all x ∈ Z and(ii) d G ( y ) > (cid:0) + ε (cid:1) z for all y ∈ W .Assuming(iii) There is an M ∈ N and with δ ≤ ε we have M ≤ d H ( x ) ≤ M (1 + δ ) ∀ x ∈ Z, and(iv) d H ( y ) = 1 ∀ y ∈ W .Then there is an embedding of H into G .Proof. We show that the conditions of Lemma 7 are satisfied.Let X ⊆ Z , Y ⊆ W . We have five cases to consider depending on thesize of X and Y .In all cases we will use the obvious inequality M z ≤ n , as d H ( X ) = d H ( Y ). For sake of simplicity, we use e ( X, Y ) = e G ( X, Y ).(a) | X | ≤ z δ ) and | Y | ≤ n .We have d H ( X ) ≤ M (1+ δ ) | X | ≤ M (1+ δ ) z δ ) = M z ≤ n ≤ | Y | = d H (cid:0) Y (cid:1) . (5)6b) | X | ≤ z δ ) and | Y | > n .Let φ = | Y | n − , so | Y | = (cid:0) + φ (cid:1) n . Obviously, 0 ≤ φ ≤ .Therefore, d H ( Y ) = | Y | = (cid:0) − φ (cid:1) n .Since d H ( X ) ≤ n , as we have seen above, furthermore, e ( X, Y ) ≥ ( ε + φ ) n | X | ≥ ( ε + φ ) n, (6)we obtain d H ( X ) ≤ d H ( Y ) + e G ( X, Y ).(c) z ≥ | X | > z δ ) and | Y | ≤ n .Let ψ = | X | z − δ ) , hence, | X | = (cid:16) δ ) + ψ (cid:17) z . Let ψ = δ δ ) = − δ ) , so ψ ≤ ψ . This means that | X | = (cid:0) − ψ + ψ (cid:1) z .As 0 < δ ≤ ε , we have ψ < δ ≤ ε .Let φ = − | Y | n , so | Y | = (cid:0) − φ (cid:1) n . As | Y | ≤ n , this gives 0 ≤ φ ≤ .(1) d H ( Y ) = | Y | = n (cid:0) + φ (cid:1) (2) As above, d H ( X ) ≤ M (1 + δ ) | X | = M z (1 + δ ) (cid:16) δ ) + ψ (cid:17) ≤ n (1 + δ ) (cid:16) δ ) + ψ (cid:17) .(3) We claim that e ( X, Y ) ≥ | Y | (cid:0) ε − ψ + ψ (cid:1) z . Indeed, the numberof neighbours of a vertex y ∈ Y in X is at least (cid:0) ε + ψ − ψ (cid:1) z ,considering the degree bounds of W in H .We show d H ( X ) ≤ e ( X, Y ) + d H ( Y ).It follows from n (1 + δ ) (cid:18) δ ) + ψ (cid:19) ≤ n (cid:18) − φ (cid:19) (cid:16) ε − ψ + ψ (cid:17) z + n (cid:18)
12 + φ (cid:19) . (7)This is equivalent to ψ + δψ ≤ z (cid:18) − φ (cid:19) (cid:16) ε ψ − ψ (cid:17) + φ. (8)The left hand side of (8) is at most ψ + δψ ≤ δ + δ ≤ δ , as δ ≤ ε ≤ .7f φ > δ , (8) holds, since ε + ψ − ψ ≥
0, using ψ ≤ ε .Otherwise, if φ ≤ δ , the right hand side of (8) is z (cid:18) − φ (cid:19) (cid:16) ε ψ − ψ (cid:17) ≥ (cid:18) − δ (cid:19) (cid:18) ε − δ (cid:19) z. (9)We also have ε δ − δε − δ > δ, (10)since ε + 2 δ − δε > ε − ε > ε > δ, (11)using δ ≤ ε .This completes the proof of this case.(d) | X | > z and | Y | ≤ n .We have(1) d H ( X ) = d H ( Z ) − d H ( X ) = n − d H ( X ) ≤ n − M | X | , (2) d H ( Y ) = n − | Y | and(3) e ( X, Y ) ≥ | Y | (cid:0) | X | − z + εz (cid:1) , using to the degree bound on Y .All we have to check is whether n − M | X | ≤ n − | Y | + | Y | (cid:16) | X | − z εz (cid:17) (12)It is equivalent to0 ≤ | Y | (cid:16) | X | − z εz − (cid:17) + M ( z − | X | ) (13)(13) has to be true for any Y and M . Specially, with | Y | = M = 1,(13) has the following form:0 ≤ | X | − z εz − z − | X | = z εz − . (14)(14) is true if z ≥ z = 1, then Z = { v } is only one vertex, which is connected to eachvertex in W . In this case, Lemma 12 is obviously true.8e) | X | > z δ ) and | Y | > n .Let ψ = | X | z − δ ) , hence, | X | = z (cid:16) δ ) + ψ (cid:17) . Let ψ = δ δ ) , as itwas defined in Case (c). Again, ψ ≤ δ . We have 0 ≤ ψ ≤ + ψ ≤ δ .Let φ = | Y | n − , hence, | Y | = n (cid:0) + φ (cid:1) .We have(1) d H ( X ) ≤ zM (1 + δ ) (cid:16) δ ) + ψ (cid:17) ≤ n (1 + δ ) (cid:16) δ ) + ψ (cid:17) , (2) d H ( Y ) = n (cid:0) − φ (cid:1) and(3) e ( X, Y ) ≥ z (cid:16) δ ) + ψ (cid:17) ( φ + ε ) n. From the above it is sufficient to show that n (1 + δ ) (cid:18) δ ) + ψ (cid:19) ≤ n (cid:18) − φ (cid:19) + z (cid:18) δ ) + ψ (cid:19) ( φ + ε ) n. (15)It is equivalent to ψ (1 + δ ) ≤ − φ + z (cid:18) δ ) + ψ (cid:19) ( φ + ε ) . (16)Using ψ ≤ δ and δ ≤ ε , the left hand side of (16) is at most1 + δ δ ) = 12 + δ + δ ≤
12 + ε
10 + ε ≤
12 + 110 = 35 , (17)as ε ≤ .The right hand side of (16) is φ z − δ )2(1 + δ ) + z δ ) ε + zψ ( φ + ε ) (18)The first and the last term of (18) is always positive. (We use that z > z δ ) ε .It is enough to show that 35 ≤ z δ ) ε. (19)9his is true indeed, since ε > z and δ ≤ ε ≤ .We have proved what was desired. Proof. (Theorem 8) First, form a partition C , C , . . . , C k of S in the graph H . For i > u ∈ C i if and only if ε n log n · δ ) i − ≥ d H ( u ) > ε n log n · δ ) i with δ = ε . Let C be the class of the isolated points in S . Notethat the number of partition classes, k is log δ n = log ε n = log n log ( ε ) = c log n .Now, we embed the partition of S into A . Take a random orderingof the vertices in A . The first | C | vertices of A form A , the vertices | C | +1 , . . . , | C | + | C | form A etc., while C maps to the last | C | vertices.Obviously, C can be always embedded.We say that a partition class C i is small if | C i | ≤ ε log n .We claim that the total size of the neighbourhood in B of small classesis at most εn .The size of the neighbourhood of C i is at most ε n log n · δ ) i − · ε log n. (20)If we sum up, we have that the total size of the neighbourhood of smallclasses is at most k X i =1 ε n log n · δ ) i − · ε log n = 425 ε n k − X i =0 δ ) i ≤≤ ε n δδ ≤ ε n / ε/ ≤ εn . (21)The vertices of the small classes can be dealt with using a greedy method:if v i is in a small class, choose randomly d H ( v i ) of its neighbours, and fixthese edges. After we are ready with them, the degrees of the vertices of B are still larger than (cid:0) + ε (cid:1) n .Continue with the large classes. Reindex the large classes D , . . . , D ℓ and form a random partition E , . . . , E ℓ of the unused vertices in B suchthat | E i | = P u ∈ D i d H ( u ). We will consider the pairs ( D i , E i ).We will show that the conditions of Lemma 12 are satisfied for ( D i , E i ).10or this, we will use the Azuma–Hoeffding inequality.We have to show that for any i every vertex y ∈ E i has at least (cid:0) + ε (cid:1) z neighbours in D i and every vertex x ∈ D i has at least (cid:0) + ε (cid:1) z in E i .Then we apply Lemma 12 with ε instead of ε , and we have an embed-ding in each pair ( D i , E i ), which gives an embedding of H into G .Let | D i | = z . We know z > ε log n , as D i is large.Build a martingale Z = Z , Z , . . . , Z z . Consider a random ordering v , . . . , v z of the vertices in Z . Let X i = 1 if v i is a neighbour of y , otherwise,let X i = 0. Let Z i = i P j =1 X j , and let Z = 0. This chain Z i is a martingaleindeed with martingale differences X i ≤
1, which is not hard to verify.According to the Azuma–Hoeffding inequality [1, 10] we have the fol-lowing lemma:
Lemma 13 (Azuma [1]) . If Z is a martingale with martingale differences , then for any j and t the following holds: P ( Z j ≥ E Z j − t ) ≥ − e − t j . (22)The conditional expected value E ( Z z |Z ) is E Z z = (cid:0) + ε (cid:1) z .Lemma 13 shows that P (cid:18) Z z ≥ (cid:18)
12 + ε (cid:19) z (cid:19) ≥ − e − ε z / z = 1 − e − ε z/ . (23)We say that a vertex v ∈ E i is bad , if it has less than (cid:0) + ε (cid:1) z neighboursin D i . Lemma 13 means that a vertex v is bad with probability at most e − ε z/ . As we have n vertices in B , the probability of the event that anyvertex is C -bad is less than n · e − ε z/ < n , (24)as z > ε log n .Then we have that with probability 1 − n no vertex in E i is bad. Thus,Condition (ii) of Lemma 12 is satisfied with probability 1 for any pair( D i , E i ).Using Lemma 13, we can also show that each x ∈ D i has at least (cid:0) + ε (cid:1) | E i | neighbours in E i with probability 1.Thus, the conditions of Lemma 12 are satisfied, and we can embed H into G . The proof of Theorem 8 is finished.11 cknowledgements I would like to thank my supervisor, B´ela Csaba his patient help,without whom this paper would not have been written. I also expressmy gratitude to P´eter L. Erd˝os and to P´eter Hajnal thoroughly forreviewing and correcting the paper. This work was supported byT ´AMOP-4.2.2.B-15/1/KONV-2015-0006 eferences [1] K. Azuma, Weighted sums of certain dependent random variables , To-hoku Mathematical Journal (1967), no. 3, 357–367.[2] A. Busch, M. Ferrara, M. Jacobson, H. Kaul, S. Hartke, and D. West, Packing of graphic n -tuples , Journal of Graph Theory (2012), no. 1,29–39.[3] P. A. Catlin, Subgraphs of graphs , Discrete Mathematics (1974),225–233.[4] J. Diemunsch, M. Ferrara, S. Jahanbekam, and J. M. Shook, Extremaltheorems for degree sequence packing and the 2-color discrete tomogra-phy problem , SIAM Journal of Discrete Mathematics (2015, in press).[5] D. Gale,
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