The Choice Number of Random Bipartite Graphs

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1 Aals of Combiatorics 2 (998) **-** The Choice umber of Raom Bipartite Graphs oga Alo a Michael Krivelevich 2 Departmet of Mathematics, Raymo a Beverly Sackler Faculty of Exact Scieces, Tel Aviv Uiversity, Tel Aviv, Israel a Istitute for Avace Stuy, Priceto, 08540, USA oga@math.tau.ac.il 2 School of Mathematics, Istitute for Avace Stuy, Priceto, 08540, USA mkrivel@math.ias.eu Receive September 7, 998 AMS Subect Classificatio: 05C5, 05C35 Abstract. A raom bipartite graph G p is obtaie by takig two isoit subsets of vertices A a B of cariality each, a by coectig each pair of vertices a! A a b! B by a ege raomly a iepeetly with probability p " p. We show that the choice umber of G p is, almost surely, # o log 2 p for all values of the ege probability p " p, where the o term tes to 0 as p tes to ifiity. Keywors: ******. Itrouctio A graph G $&% V' E( is calle k-choosable for a iteger k ) 0 if, for every family of color lists*+$-, S% v(/. Z : v 0 V % G(2 satisfyig S% v(3$ k for every v 0 V, there exists a choice fuctio f : V 4 Z such that f % v(50 S% v( for all v 0 V, a also f % u(76 $ f % v( for every ege e $-% u' v(80 E % G(. The choice umber ch% G( of G is the miimal iteger k for which G is k-choosable. The cocept of choosability was itrouce by Vizig i 976 [6] a iepeetly by Erös, Rubi a Taylor i 979 [5]. Although the choice umber is a straight-forwar geeralizatio of the more familiar otio of the chromatic umber, after twety years of research, it appears to be a much more complicate quatity, a much less is kow about it. The reaer may cosult the survey paper of the first author [2] for a iscussio of various problems a results o choosability. Our iterest i choosability questios for bipartite graphs is stimulate, i particular, by the fact that bipartite graphs provie a staar example of a family of graphs for 9 Research supporte i part by a USA Israeli BSF grat, a grat from the Israel Sciece Fouatio, a Sloa Fouatio grat o a a State of ew ersey grat. Research supporte by a IAS/DIMACS Postoctoral Fellowship.

2 G 2. Alo a M. Krivelevich which the choice umber ca be much higher tha the chromatic umber. (It follows immeiately from the efiitio of the choice umber that ch% G(;: χ% G( for every graph G). Iee, Erös, Rubi a Taylor otice i their origial paper that the choice umber of the complete bipartite graph K < is % = o% (>( log 2. The (relatively simple) proof of this statemet cotais several useful ieas, some of which are applie i the preset ote. The mai goal of this ote is to etermie the asymptotic behavior of the choice umber of raom bipartite graphs. Formally, the raom bipartite graph G% m' ' p( is the probability space whose poits are bipartite graphs o a fixe set of m = labele vertices, partitioe ito two color classes A a B of carialities A?$ m, B?$, respectively. Each pair of vertices a 0 A a b 0 B form a ege raomly a iepeetly with probability p $ p% m' (. The color classes A a B form iepeet sets. By the term the raom bipartite graph we mea a raom poit chose i this probability space. I this ote we cofie ourselves to the case where the color classes, the correspoig moel is eote by A a B are of equal cariality A$& B@$ G% ' ' p(. We use the usual otatioal covetios of the theory of raom graphs. As is usually the case i this subect, we are itereste i the behavior of various parameters as tes to ifiity. We say that a graph property A hols almost surely ( a.s. for short) if the probability that G% ' ' p( satisfies A tes to as tes to ifiity. ote that $ p is the expecte egree of each vertex of G% ' ' p(. The problem of etermiig the asymptotic value of the choice umber of raom bipartite graphs has alreay bee aresse by Erös, Rubi a Taylor [5]. They cosiere the moel G% ' ' A 2( a prove that a.s. loga log6 B ch% G% ' ' A 2(C(DB 3logA log6. I this paper we etermie the typical asymptotic value of ch% G% ' ' p(c( for all values of the ege probability p $ p% ( ow to p : CA for some costat C ) 0. We prove the followig theorem. Theorem.. There exists a absolute costat 0 such that, if the ege probability p $ p% ( satisfies p ) 0, the a.s. log 2 % p(fe 4log 2 log 2 % p(hg ch% G% ' ' p(>( 5log log 2 % p(i= 2 % p( log 2 log 2 log 2 % p( log 2 log 2 % p( Thus, the choice umber of G% ' ' p( is almost surely % = o% (C( log 2 for all values of p% (. We o ot make here ay serious attempt to optimize the error terms i Theorem., our mai task is to fi a asymptotic formula for the mai term. The rest of the paper is orgaize as follows. I the ext sectio we preset some properties of the ege istributio i raom bipartite graphs require for the subsequet proofs. The proof of the lower bou is give i Sectio 3. Sectio 4 is evote to the proof of the upper bou. Sectio 5, the fial sectio of the paper, cotais several cocluig remarks a a iscussio of relevat ope problems. Throughout the paper, all logarithms are i base 2. As metioe above we eote by $ p the expecte vertex egree i G% ' ' p( a assume, wheever eee, (a hece also ) is sufficietly large. We omit routiely floor a ceilig sigs wheever these are ot crucial to simplify the presetatio.

3 O V Choosability of Raom Bipartite Graphs 3 2. Prelimiaries I this sectio we prove two techical propositios about the ege istributio i bipartite raom graphs. As show i the ext two sectios, these propositios are essetially the oly properties of raom bipartite graphs that are eee to prove our result. Propositio 2.. The raom bipartite graph G% ' ' p( has a.s. the followig property: For every two subsets X. A, Y. B of carialities X K'L Y M: 2logA, there exists a ege e 0 E % G( coectig X a Y. Proof. The probability that there exists a pair X ' Y violatig the assertio of the propositio is at most 2log 2 2log 2 % E p(qp R GTS e 2log O 2 eu 2plog V 2log GXW 2 eu 2log 2 Y 2log $ o% ( Propositio 2.2. For every fixe costat C ) 0 the raom bipartite graph G% ' ' p( has a.s. the followig property: For every two subsets X. A, Y. B, satisfyig X Z'L Y 3G CA, the spae subgraph G[ X \ Y] of G o X \ Y has at most %C X =^Y ( 3logA log log eges. Proof. The result is trivial for $ Ω% (, hece, we may a will assume $ o% (. Deote ε $ ε % ($ 3logA loglog. The probability of existece of subsets X, Y violatig the propositio ca be boue from above Author: somethig missig here? **** by i< _ i O G 2 c ic c C G 2 c ic c C G 2 c ic c C $ 2 S _ $ 2 S _ e e i % i = ( ε O p` ia b ε i O e i % i = ( ε O p` ia b ε ei ` ia b ε % i = ( ε O e p ε O ε V e p ε O ε 2 % e p( 2 % e p( ε U 2 ε ε p` ia b ε

4 $ 4. Alo a M. Krivelevich G 2 e _ G 2 _ e 4 2 % e p( ε U 2 3 U o` b f % p( ε U 2 0g 9 O Deote the -th summa of the last sum by s. The, if Gh% A ( 2, we obtai s Gh% A ( ε ` U 3bi 2 $ o% A (. If :-% A ( 2, the s G% CA (3% C ε U 2 A 0g 9 ( ` b k 2 o% A (, thus, showig that the last sum tes to 0 as tes to ifiity. 3. Proof of the Lower Bou To prove the lower bou of Theorem., we argue etermiistically that every bipartite graph G $% A \ B' E( with A3$l Bm$, satisfyig the assertio of Propositio 2. for some value of the parameter, also satisfies ch% G(: log E 4loglog. Agai, we assume the parameter (a hece ) to be large eough. Let t% k( eote the miimal umber of eges i a k-uiform o-2-colorable hypergraph H. A tight coectio betwee the problem of etermiig t % k( a choosability questios for bipartite graphs has bee expose alreay i the origial paper of Erös, Rubi a Taylor [5]. As show by Erös [4], t % k(go% = o% (>(m% el2a 4( 2 k k 2. Give a, efie a iteger k $ k% ' ( by k $ max p i : 2ilog Grq t % i(?sutv It is easy to see that k : log Ew% = o% (C( 3loglog ) log E 4loglog. Therefore, to prove the esire lower bou o ch% G(, it suffices to prove that ch% G(x) k. Let us eote t $ t % k( to simplify the otatio. Give a bipartite graph G $&% A \ B' E(, we partitio the color class A ito t color classes A $ A \ \ A C> t so that A i I:zy A t{, G i G t. Similarly, we partitio B $ B \ \ B C> t with B i :Ty A t{. Let H $z% V' F( be a k-uiform o-2-colorable hypergraph with t eges. Deote F $&, S ' ' S >> t 2. We view the vertices of V % H( as colors, while the subsets S i will be assige to vertices of G as color lists. For each G i G t, every vertex v 0 A i \ B i obtais S i as its list of colors. We claim that the assertio of Propositio 2. alog with the o-2-colorability of H guaratee that G caot be properly colore by assigig each vertex a color from its list. Iee, let f : A \ B 4 V % H( satisfy f % v(}0 S% v( for every v 0 V % G(. Let C be the subset of V % H(, forme by all the colors chose by f at least 2logA times o A. For every G i G t, the list S i of cariality S i?$ k is assige to all vertices a 0 A i. As A i ~:y A t{ : 2klogA, we obtai that at least oe color from S i is chose at least 2logA times o A i. Thus, C itersects every ege S i of H. ow, as H is o-2-colorable, there exists a ege S i0 0 F, satisfyig S i0 C (otherwise, the partitio V $ C \ % V ƒ C ( forms a 2- colorig of H). The color list S i0 is assige to all vertices b 0 B i0, hece, there ex- 2logA times o ists a color c 0 S i0, chose by f at least B i0 A S i0 :Ty A t{?a k : B i0. Let X $, a 0 A : f % a($ c 2, Y $, b 0 B : f % b(8$ c 2. As c 0 S i0 C, we have

5 G G ' Choosability of Raom Bipartite Graphs 5 X m: 2logA. Also, Y m: 2logA. But the accorig to the claim of Propositio 2., there exists a ege of G coectig X a Y, thus showig that f oes ot form a proper colorig of G. This coclues the proof of the lower bou of Theorem.. 4. Proof of the Upper Bou The proof here is also etermiistic. We assume a bipartite graph G $v% A \ B' E( has the property give by Propositio 2.2 a show that every graph havig this property has its choice umber boue from above by the upper bou of Theorem.. Let us itrouce the followig otatio. Defie ε 2 i_ 0 l i O ε $ ε % ( $ ε 2 $ ε 2 % ( $ ε 3 $ ε 3 % ( $ 3log loglog ' 4log loglog 5log logloglog loglog l $ l% ( $ y log = ε 3 { Give a family * $o, S% v( : v 0 A \ B2 satisfyig S% v(m$ l for every v 0 A \ B, our aim is to prove the existece of a choice fuctio f. Deote by S $-ˆ v AŠ B S% v( the uio of all colors i all lists. Partitio S raomly ito two parts S A a S B by puttig each color c 0 S ito S A or S B iepeetly a with probability A 2. Ieally, we woul like to use colors from S A to color vertices from A a those from S B to color vertices from B. This strategy woul elimiate ay possible color coflict. The problem is that we caot guaratee that every vertex from A \ B obtais at least oe eligible color uer such a partitio. However, we will be able to show that with positive probability the umber of vertices that obtai oly few colors is quite small. We call a vertex a 0 A poor if S% a(~ S A ŒB ε 2. Similarly, a vertex b 0 B is poor if S% b(~ S B MB ε 2. Let T 0 eote the set of all poor vertices. The probability that a 0 A is poor is at most l 2U G 2U l ε 2 2U ε2 ε 3 el ε 2 ε 2 O l ε 2O 2U ε 3 ε ε 2 % loglog( 2 $ 2U ε 3 a logε 2U ε 2 logloglog B 2 The same argumet shows that, for every b 0 B, the probability that B is poor is less tha A % 2(. We coclue that with positive probability, T 0 LG A. Let us fix a partitio S $ S A \ S B, for which iee T 0 QG A. ow, we fi a (small) subset T. A \ B, icluig T 0, such that every vertex of G outsie T has less tha ε 2 eighbors isie T. To fi such a subset, we start with T $ T 0, a as log as there exists a vertex v 0 % A \ B( ƒ T havig at least ε 2 eighbors i T, we a v to T. This process stops with T ŒG 5A, otherwise, we woul obtai

6 6. Alo a M. Krivelevich a subset T of size T $ 5A cotaiig at least % 4A ( ε 2 $ 6logAI% loglog( eges, thus cotraictig the assertio of Propositio 2.2. We claim that we ca fi a proper colorig f by startig from choosig colors for vertices of T, a the by usig colors from S A to color vertices from A ƒ T a colors from S B to color those from B ƒ T. Accorig to Propositio 2.2, the spae subgraph G[ T] has a vertex of egree at most 2ε i each of its subgraphs. This shows that G[ T] is 2ε -egeerate a thus % 2ε = ( -choosable (see, e.g., [2] for a iscussio of the coectio betwee egeeracy a choosability a for a very simple proof of the above statemet). Thus, we ca use the origial lists of colors, S% v( : v 0 T 2 to fi colors for all vertices from T. ext, for every a 0 A ƒ T, we elete from S% a(i S A the colors chose for eighbors of a i T, a similarly, for every b 0 B ƒ T, we elete from S% b(i S B the colors chose for eighbors of b i T. As all poor vertices fall isie T a every vertex outsie T has less tha ε 2 eighbors isie T, eve after this eletio each vertex i % A \ B( ƒ T still has at least oe eligible color. We complete the choice of colors by choosig a arbitrary remaiig color i S% a(i S A for each a 0 A a a arbitrary remaiig color i S% b(i S B for each b 0 B. This completes the proof of the upper bou of Theorem.. 5. Cocluig Remarks The proof i Sectio 3 shows that ay bipartite graph with sufficietly strog expasio properties has a large choice umber. More precisely, if G is a bipartite graph with vertices i each of its two color classes A a B, a there is at least oe ege betwee ay subset of cariality A x of A a ay subset of cariality A x of B, the the choice umber of G is at least % = o% (>( log 2 x, where the o% ( -term tes to 0 as x tes to ifiity. By the kow relatio betwee the eigevalues of the aacecy matrix of a graph a its expasio properties this implies that the choice umber of ay -regular bipartite graph G i which the absolute value of every eigevalue besies the largest a the smallest is at most λ, satisfies ch% G(Ž:h% = o% (>( log 2 % A λ( This is because, i each such graph, there is a ege betwee ay two subsets of the two color classes provie each subset is of cariality at least λa. The argumets i Sectios 3 a 4 ca be extee to eal with the choice umbers 2. Let G r % ' p( eote the raom r-partite graph obtaie by takig r pairwise isoit sets A ' A 2 ' ' A >> r, each of cariality, a by coectig each pair of vertices i istict sets A i by a ege, raomly a iepeetly, with probability p $ p% (. We ca show that, for every fixe r : 2 a every p $ p% (, the choice umber of G r % ' p( is almost surely % = o% (>( log% p(ca log % ra % r E (>(, where the o% ( term tes to 0 as p tes to ifiity. The proof is similar to the oe give here for the case r $ 2. We omit the etails. We ote that the choice umber of the complete r-partite graph with vertices i each color class is Θ% r log( for all a r, as prove i [], a the choice umber of the usual raom graph G% ' p% (>( is almost surely Θ% p% (CA log% p% (>(>( wheever 2 B p% (8B 9A 0, as prove i [3]. I [2], it is prove that the choice umber of ay graph with average egree at least of raom r-partite graphs for r :

7 Choosability of Raom Bipartite Graphs 7 is at least Ω% loga loglog(. It seems plausible that the loglog term ca be omitte, but at the momet this remais ope. If true, this woul, of course, be tight up to a costat factor (sice, for example, ch% K < (/$ % = o% (C( log 2 ). It is ot ifficult to prove that the choice umber of ay bipartite graph with maximum egree is at most O% A log(, but we believe that the followig much stroger result hols. Coecture 5.. The choice umber of ay bipartite graph with maximum egree is at most O% log(. As far as we kow it is eve possible that the choice umber of each such graph is at most % = o% (>( log 2, where the o% ( term tes to 0 as tes to ifiity. As a test case, it may be iterestig to etermie or estimate the choice umber of the -cube. Refereces.. Alo, Choice umbers of graphs: a probabilistic approach, Combi. Probab. Comput. (992) Alo, Restricte colorigs of graphs, I: Surveys i Combiatorics 993, Loo Math. Soc., Lecture otes Series, Vol. 87, K. Walker, E., Cambrige Uiversity Press, 993, pp Alo, M. Krivelevich a B. Suakov, List colorig of raom a pseuo-raom graphs, (which oural??), to appear. 4. P. Erös, O a combiatorial problem II, Acta Math. Aca. Sci. Hugar. 5 (964) P. Erös, A.L. Rubi a H. Taylor, Choosability i graphs, I: Proc. West Coast Cof. o Combiatorics, Graph Theory a Computig, Cogressus umeratium XXVI, 979, pp V.G. Vizig, Colorig the vertices of a graph i prescribe colors, Diskret. Aaliz. o. 29, Metoy Diskret. Aal. v. Teorii Koov i Shem 0 (976) 3 0 (i Russia).