COMBINATORICA 12 (3) (1992) 295 301 COMBINATORICA Akad6miai Kiad6 - Springer-Verlag ON PACKING BIPARTITE GRAPHS PI~TER HAJNAL and MARIO SZEGEDY Received November22, 1989 G and H, two simple graphs, can be packed if G is isomorphic to a subgraph of H, the complement of H. A theorem of Catlin, Spencer and Sauer gives a sufficient condition for the existence of packing in terms of the product of the maximal degrees of G and H. We improve this theorem for bipartite graphs. Our condition involves products of a maximum degree with an average degree. Our relaxed condition still guarantees a packing of the two bipartite graphs. O. I n t r o d u c t i o n If G is a graph, t h e n V(G), E(G), D(G), ~(G), -[l(G) will d e n o t e its v e r t e x set, edge set, m a x i m a l degree, m i n i m a l degree a n d average degree. Let U be a n y s u b s e t of V(G). Let Du(G ) a n d "du(G) b e t h e m a x i m u m a n d average degrees where t h e m a x i m u m a n d average t a k e n over t h e vertices in U (the c o r r e s p o n d i n g degrees are b a s e d on t h e whole g r a p h ) . Let Gv be t h e collection of g r a p h s w i t h vertex set of size v. Let Bu,w be t h e collection of b i p a r t i t e g r a p h s w i t h two color classes of size u a n d w. If f be a 1-1 m a p from V(G) o n t o W , let G f b e t h e i m a g e of G u n d e r f , i.e. a g r a p h on t h e v e r t e x set W . Definition 0.1 (a) Let G, HEGv. A packing is a bijection f:V(H)---*V(G) such t h a t t h e edge set of G a n d H f are disjoint. (b) Let G, H E Bu,w. Let us a s s u m e t h a t G has color classes U a n d W a n d H has color classes U I a n d W I. A bipartite packing is a bijection f t h a t m a p s U I to U a n d W ~ to W such t h a t t h e edge set of G a n d H f are disjoint. P a c k i n g g r a p h s is a heavily s t u d i e d s u b j e c t in g r a p h theory. A g o o d survey of this research can b e found in [2]. Next we s u m m a r i z e t h e known results on packing. M u c h effort has been s p e n t for packing sparse g r a p h s [15], [17], [5], [6], [13], [16], [10]. A t y p i c a l t h e o r e m from this a r e a is: AMS subject classification code (1991): 05 C 70 The paper was written while the authors were graduate students at the University of Chicago and was completed while the first author was at M.I.T. The work of the first author was supported in part by the Air Force under Contract OSR-86-0076 and by DIMACS (Center for Discrete Mathematics and Theoretical Computer Science), a National Science Foundation Science and Technology Center - NSF-STC88-09648. The work of the second author was supported in part by NSF grant CCR-8706518. 296 PI~TER HAJNAL, M,~RIO SZEGEDY Theorem 0.2. (N. Sauer and J. Spencer [151) If IE(G)[, IE(H)I <_v-2 (where IV(G)I= [V(H)] =v) then G and H can be packed. One can extend packing to packing several graphs. We just refer the reader to [11], and we mention a nice conjecture from this paper. Conjecture 0.3. (A. Gys163 and J. Lehel [11]) Let T k be any tree with vertex set of size k (k = 1,...,n). Then there is a packing of T1,T2,...,Tn into the complete graph on n vertices. The following few theorems give sufficient conditions on the number of edges for the existence of a packing. Theorem 0.4. (B. Bollobs and S.E. Eldridge [4]) If IE(G)[ + IE(H)[ < [~(v - 1)] (where IV(G)[ = IV(H)[ =v) then there is a packing of G and H. For improvements (but still with a linear upper bound in the condition on the sum of the number of edges) see [4]. Theorem 0.5. (B. Bollobs and S.E. Eldridge [4]) (i) If G, H eGv and IE(G)[[E(H)I < (v2) then G and H can be packed. (ii) If G , H 6 Bu,w and [E(G)[[E(H)I < uw, then G and g can be packed as bipartite graphs. Theorem 0.6. (B. Bollobs and S.E. Eldridge [4]) 1 3 (i) If G, H e G v , IE(H)I< ~ and [E( G)[< rhv2 then G and H can be packed. 1 3 (ii) If G, g 6 Bu,u, [E(H)I < ~ and [E(G)[ < 1-5u~ then G and H can be packed as bipartite graphs. For us the most important sufficient conditions will be the following ones on the maximal degrees. Theorem 0.7. (Conditions on the maximal degree [15],[7]) (i) If G, H E G v and D ( G ) D ( H ) < v / 2 then G and H can be packed. (ii) If A , B E Bu,w and D u ( A ) D w ( B ) + D w ( A ) D u ( B ) < u, then A and B can be packed as bipartite graphs. In the last two statements the bounds in the conditions are tight (up to negligible factors). For Theorem 0.7.(ii) (the bipartite case) this can be easily shown using the probabilistic method in [15]. For Theorem 0.7.(i) there is an easy construction [4] showing that one cannot improve the condition with more than a factor of 2. That example suggests the following conjecture. Conjecture 0.8. (B. Bollobgs and S.E. Eldridge [4]) Let G and H be two graph on a vertex set of size v. If (D(G) + 1)(D(H) + 1) <_v + 1 then there is a packing of G and H. The condition in Theorem 0.7.(ii) restricts the product of the maximum degrees of G and H. Our improvement comes from relaxing one of the terms to average degree. 297 ON PACKING B I P A R T I T E G R A P H S Theorem 0.9. Let G, H E Gu,w. Assume that (a) u<w<2u, -du,( G)Dw( H) < u/IO0, -du( H)Dw,( G) < u/IO0, Du(G),Du,(H ) <_u/lOOOlogu. Then G and H can be packed. (b) (c) (d) This research was motivated by questions on decision tree complexity. For an application of this theorem in this direction see [12]. 1. T h e improved packing theorem for bipartite g r a p h s In this section we prove the result stated in the introduction. First, we review Catlin's idea. Given G,H E Gu,w with color classes U,W and U I, W I, resp. We want to find a sufficient condition for existence a packing. We take an arbitrary bijection f : U I --* U. Define a bipartite graph between W and W I based on whether two nodes can be identified or not. Now the problem is simply finding a matching in this auxiliary graph. DeAn|tion 1.1 Let G, H C Gu,w. Let U, W, U t and W ~ be the corresponding color classes. Given f , a bijection U ~ --* U, we, define a bipartite graph B f with color classes W and W I. We make x E W and y E W t adjacent iff x and y can be identified, i.e., the neighborhoods of x in G and of y in H f are disjoint subsets of U. Now it is easy to show that if G and H satisfy the condition of Theorem 0.7.(ii) then for any bijection f B f satisfies the condition of K6nig's theorem (see e.g. [14], Chap. 7, prob. 4.) and therefore possesses a perfect matching along which we can map W t to W to obtain a packing. It is worth to state this fact as a separate lemma. Lemma 1.2. (i) Let GeBu,u. If ~u(G),~w(G)>_u/2 then G has a perfect matching. (ii) Let G E Bu,u. If 6u (G ) + ~w (G ) >__u then G has a perfect matching. The proof of our result is probabilistic. Our goal is to show that there exists a bijection f : U - ~ U I such that BI has a perfect matching. We are going to show that this is true for a random bijection. Let dl .... ,du be all the degrees in U, and let e l , . . . ,eu be all the degrees in U ~. LemmA 1.3. Let f : U-~ U I be a random bijection, all bijections being equally likely. Pr~ hasperfeetmatching)~l-wPr~176 \iES where S is a random subset ofU I of size Dw(G), M1 such subsets being equally likely, and R is a random subset of U of size Dw,(G), a11 such subsets being equally likely. Proof. We are interested in the event E = B f has a perfect m a t c h i n g . 298 PETER HAJNAL, MARI() SZEGEDY By L e m m a 1.2 the following event is a subset of E. W F = Each node of Bf has degree at least 2 " One elementary bad event is Fz = x has degree in Bf less than w ( f o r x E W O W ' ) . Using this notation E ~_ F = f~ - U z e w u w , Fx. Thus Prob(E) > 1 - E Prob(Fx). xEWUWt For x E W, Fz is exactly the event that the image f(N(x)) of N(x) (f(N(x))C U r) has a neighborhood in W t of size more than ~y. The event that the sum of the degrees in f(N(x)) is at least ~ is a superset of Fx. If x e W then f(N(x)) is a random set of size IN(x)l and its size is at most Dw(G ). This completes the proof. | Our conditions on G and H are symmetric. So it is enough to show that Prob di >- < 2w Ven R is a random subset of U I. There are different models for random sets. In our case R is a random set of a given size. Another model is that each element of our universe will be in the set with a given probability. This model is more convenient. It is well-known in the theory of random graphs [3] that by choosing the right parameters the two models yield basically the same theorems. So our next step is to change to the second model. For this we need some inequality for Bernoulli random variables. L e m m a 1.4. (Chernoff [8]) Let X 1 , X 2 , . . . , X N be independent 0-1 random variables such that Prob(Xi= 1 ) = p . Ifra>_Np is an integer then Prob Xi >_m << e x p ( m - Np). An easy consequence of this is the following. L e m m a 1.5. ([9], [1]) Let XI,X2,...,X N be independent 0-1 random variables such that Prob(Xi = 1) = p . Then for every 0 </3 < 1, (i) Prob(~iN1xi<<[(1-13)NpJ)<exp ( - ~ - ~ ) (ii) P ob <_ t(1 + 9)Npj) _<exp (- And now let us see the reduction. . ON PACKING BIPARTITE G R A P H S 299 L e m m a 1.6. Let X l , . . . , Xu be independent random variables such that Prob(Xi = d i ) = P > 2 DW(G)" " and Prob(X i = O ) = l - p . u Let A be a random subset of 1 , 2 , . . . , u of size Dw(G ). Then Pr~ <2Pr~ Proof. Let Ai be a random subset of 1,2,..., u of size i, with all i-subsets of 1,2,..., u (~je~ dj > ~)." being equally likely. Let Pi = Prob Then Prob Xi > = \i=0 Then P0 -< P1 -<...-< Pu- N pk(1 -- p)u-kp k k=l L89 <k< L~pJ 1 1 >]PL89 > ]PDw(G) = ~Prob di > 9 | So at this point using the notation of the previous lemma, we will give an upper bound on Prob(~U=l Xi>_ ~). Let us fix the value of p to be 10 DW(G).'" Notice that the conditions of Theou rem 0.9 imply w >> E( Eu -~ Xi) = E 10 D W(uG)di = IODw(G)~IU,. i=1 i So we need an upper bound on the probability that a sum of independent random variables is much greater than their expected sum. The Chernoff bound is that kind of result, but it is about Bernoulli random variables. We use the method of the proof of Chernoff's theorem to get the desired upper bound. For that we need the notion of characteristic function. Definition 1.7 Let X be a random variable. Its characteristic function is etX, a random variable depending on the real parameter t. The following lemma shows an important property of the characteristic function. Lemma 1.8. Let X1,...,XN be independent random variables. Then E etXi = 1-[ E etXi . i=l Now we have everything required to prove the last lemma that we need. 300 PETER HAJNAL, MARI() SZEGEDY Lemma 1.9. Let 0 < dl, d2,..., du <_L = u be integers and define d by ~i=1 u di = -du. Let X1, X2,..., Xu be independent random variables such that Prob(Xi = di) = p and Prob(X i =0) = 1 - p . Then Prob (EiU_l Xi > 10pdu) < 1 Proof. For all positive t Let us compute E(e~iX~t). dit) : H ( 1 - p ( 1 - e d i t ) ) . i An easy calculation shows that this product is maximal if all di's are 0 or L, the : H (1-p+pe i maximal possible value of them. So T < ( 1 - p ( 1 - (1 + 2 L t ) ) ) ~ < (1 + 2pLt)~ < e2p-due, E(eE, X#) <_( 1 - p ( 1 - e Lt ) ) ~u assuming that Lt <_1. Using Markov's inequality (V ) e2pdute-Spdut Fixing the value of t to be 1/L our bounds are still true and we obtain the desired upper bound. | We obtain the promised packing theorem (Theorem 0.9) as a corollary. Proof of Theorem {}.9, Applying Lernma 1.9, Lemma 1.6 and Lemma 1.3 we obtain that for a random f that B f has a perfect matehing with positive probability . This proves that there exists a concrete bijection f such that the corresponding B f has a perfect matching. This perfect matching is an identification of W and W r, which together with f gives us a packing. | Our proof heavily uses the fact that we are working with bipartite graphs. It is an interesting open question whether one can extend our result to the case of general graphs. Acknowledgement. The authors are grateful to L~szl6 Babai for helpful discussions. ON PACKINGBIPARTITEGRAPHS 301 References [1] D. ANGLUIN, and L. G. VALIANT: Fast probabilistic algorithms for Hamiltonian circuits and matehings. Journal of Computer and System Sciences 19 (1979), 155 193. [2] B, BOLLOBXS: Extremal Graph theory, Academic Press, London, 1978. [31 B. BOLLOBXS: Random Graphs, Academic Press, London. 1985. [4] B, BOLLOBAS, and S. E. 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