A Brooks Theorem for Triangle-Free Graphs

A Bro oks’ Theorem for T riangle-F ree Graphs Mohammad Shoaib Jamall ⋆ Department of Mathematics, UC San Diego mjamall@ma th.ucsd.ed u Abstract. Let G be a triangle-free graph with maximum degree ∆ ( G ). W e sho w that the c hromatic num ber χ ( G ) is less than 67(1 + o (1)) ∆/ log ∆ . 1 In tro duction A pr op er vertex c oloring of a graph is an assignment of colors to all vertices suc h that a dj acent v er tices ha ve distinct colors. The ch r omatic n umb er χ ( G ) of a graph G is the minimu m num b er of colors required for a prop er vertex coloring . Finding the chromatic num b er o f a graph is NP-Har d [1 0]. Approximating it to within a p olynomial ratio is also hard [1 5]. F o r gene r al gra phs, ∆ ( G ) + 1 is a trivial upp er b ound. Bro oks ’ Theorem [7] shows that χ ( G ) can be ∆ ( G ) + 1 only if G has a comp onent which is either a complete subgraph or an odd cy cle. A natural question is : can this b ound be improv ed for g raphs without large complete subgra phs? In 1968, Vizing [22] had asked what the best p ossible upper b ound for the c hroma tic num b er of a triangle-free graph was. Boro din and Ko sto chk a [6], Catalin [8 ], and Lawrence [19] indep endently made pr ogress in this direction; they sho wed that for a K 4 -free graph, χ ( G ) ≤ 3( ∆ ( G ) + 2) / 4. On the o ther hand, K o sto chk a and Masurov a [18], and Bollob´ as [5] separately show ed that there ar e graphs o f arbitr a rily lar g e girth (length of a shortest cycle) with χ ( G ) of order ∆ ( G ) / log ∆ ( G ). In 1995, Kim [16] pro ved that χ ( G ) ≤ (1 + o (1 )) ∆ ( G ) log ∆ ( G ) when G has girth greater than 4. Later on, Johansson [12] sho wed that χ ( G ) ≤ O ( ∆ ( G ) log ∆ ( G ) ) when G is a triang le-free gr aph(girth grea ter tha n 3). Alon, Krivelevich and Suda ko v [3], and V u [23] extended the metho d of J ohansson to prov e b ounds on the chromatic n umber for graphs in which no subgraph on the set of all neigh bo rs of a v ertex has to o many edges. Both Kim and Johansson used the so-called semi-r andom metho d to sho w that th e c hro ma tic num b er of graphs with large girth is O ( ∆ ( G ) / log ∆ ( G )). This technique, also known as the pseudo-r andom metho d , or the R¨ odl nibbl e , app ear ed fir s t in Ajtai, Ko ml´ os and Szemer´ edi [2] and was applied to problems in h ype rgraph packings, Ramsey theor y , c o lorings, and list c olorings [9,1 3,14,17,21]. In genera l, given a set S 1 , the goal is to show that ther e is a n ob ject in S 1 with a des ir ed prop erty P . This is done b y loca ting a seque nc e of non-empty subsets S 1 ⊇ · · · ⊇ S τ with S τ having pr o p e rty P . A randomized a lgorithm is applied to S t , which guarantees that S t +1 will be obtained with some non-zero(often sma ll) probability . F or upp er bounds on chromatic nu mber, the semi-r andom metho d is used to prove the e x istence of a prop er coloring with a limited n um ber of colors. In this pap er we pr ov e that the c hromatic num ber of a tria ngle-free gra ph G is less than 67(1 + o (1)) ∆/ log ∆ . As we will indicate in Section 2, o ur pr o of is derived from Kim’s pr o of o f an upper b ound ⋆ Researc h supported in part by a Reub en H. Fleet F oundation F ello wship, and a n ARCS F oundation F ello wship. to the chromatic n um ber of g raphs with girth g reater tha n 4. W e b elieve our technique is simpler tha n Johansso ns’ whic h follows a different approach to that of Kims (see [20] for a comparison of both). W e give our pr o of by analyz ing an iterative algor ithm for graph coloring. T o ana lyze this algo rithm we ident ify a collection o f rando m v aria bles. The exp ected ch anges to these ra ndom v ar iables after a round o f the a lgorithm are written in terms of the v alues o f the ra ndo m v ariables b efore the ro und. W e thus obtain a set of recurre nc e relations and prov e that our r andom v aria bles are concentrated around the solutions to the recurrence relations with some pos itive probability . W e descr ibe our algo rithm in Section 2. Section 2.1 co nt ains motiv ation, which is followed by a formal description of the algorithm in Section 2.2. W e give an outline of the analysis in Section 3. Section 4 co nt ains some useful lemmas whic h w e are used in Section 5 to giv e details of the analysis . 2 An It erativ e Algorithm for Coloring a Graph Our algorithm takes a s input a triangle-free g r aph G on n vertices, its maximum degre e ∆ , and the num b er of colors to use ∆/k where k is a p ositive num be r . It go es through rounds and ass igns co lors to more vertices each round. Initially all v ertices a re un c olor e d (no color a ssigned), at the end we hav e a pro pe r vertex coloring of G with some probability . Definition 1 . L et t b e a n atur al nu mb er. We define the fol lowing: G t The gra ph induc e d on G by the vertic es t hat ar e un c olor e d at t he b e ginning of r ound t . N t ( u ) The set of vertic es adjac ent t o vertex u in G t . That is, the set of unc olor e d neighb ors of u at t he b e ginning of r ound t . S t ( u ) The list of c olors that may b e assigne d to vertex u in r ound t , also c al le d the palette of u . F or al l u in V ( G ) , S 0 ( u ) = { 1 , . . . , ∆/k } . D t ( u, c ) The s et of vertic es adjac ent to u t hat may b e assigne d c olor c in r ound t . That is, D t ( u, c ) := { v ∈ N t ( u ) | c ∈ S t ( v ) } . It will be useful to define v a riables for the sizes of the sets S t ( u ) and D t ( u, c ). Definition 2 . s t ( u ) = | S t ( u ) | d t ( u, c ) = | D t ( u, c ) | Observe that for every round t , v er tex u in V ( G ), and colo r c in { 1 , . . . , ∆/k } , d t ( u, c ) ≤ ∆, s 0 ( u ) = ∆/ k , s t ( u ) ≤ ∆/ k . 2.1 A Sketc h of the Algo ri thm and the Ideas b e hi nd its Analysis W e say that a sequence x ( n ) is O ( f ( n )) if ther e is a p o s itive num b er M such that | x ( n ) | ≤ M | f ( n ) | . All sequences in the big-oh a re indexed by ∆ , the maximum degree of gr a ph G . Remem be r that each o ccurr ence of the big-o h comes with a distinct constant M . W e start by considering an algorithm th at colors ∆ -r egular graphs with girth greater than 4. The coloring pro duced is prop er with positive probability . 2 Let ( d t ) and ( s t ) be s equences defined recursively as d 0 := ∆ d t +1 := d t (1 − c 1 s t d t ) c 2 s 0 := ∆/k s t +1 := s t c 2 where c 1 and c 2 are constants b etw een 0 and 1 , whic h are determined by the analysis of the algorithm. Rep eat at every round t , until d t /s t < 1 / 2 Phase I - Coloring Attempt F or each v ertex u in G t : Awa ke vertex u with pro babilitit y s t /d t . If awake , assign to u a color c ho sen from S t ( u ) uniformly at random. Phase II - Conflict R esolut ion F or each v ertex u in G t : If u is awake , uncolor u if an adjacent vertex is assigned the same color. Remov e from S t ( u ), all colors assigned to adjacent vertices. S t +1 ( u ) = S t ( u ) . end rep eat Permanen tly color each v ertex u in V ( G t ) with a color pic ked independently and unifor mly at random from its palette S t ( u ). Observe that d 0 /s 0 = k and d t +1 s t +1 = d t s t − c 1 . In O ( k ) r ounds d t /s t will be less than 1 / 2, and this marks the end of the repeat-until block. The algor ithm above is derived fro m Kim [1 6]. After some mo dificatio ns, his a na lysis tells us that if graph G has g irth greater than 4 , then there ar e constants c 1 and c 2 less than 1 such that at each ro und t , ∀ u ∈ V ( G t ) , ∀ c ∈ S t ( u ) s t ( u ) = s t (1 + o (1)) , d t ( u, c ) = d t (1 + o (1)) (1) with pro bability grea ter than 0. The equations a b ove imply that after the rep eat-until blo ck, if ∆ is large enough, then with p o s itive probability s t ( u ) > 2 d t ( u, c ) for all uncolored v er tices u a nd colo r s c in their palette. Now, applying the res ult of Haxell [11], w e find that the ra ndom assig nmen t of co lors to a ll uncolo red vertices in the final step o f the algorithm gives a prop erly color e d g raph with positive probability . The proble m with cycles of l ength 4. The analysis for the ab ov e algorithm is probabilistic and prov es the pr op erty in equation (1 ) by induction, sho wing concen tra tion of the v ariables ar ound their ex pec ta tions. It fails for graphs with 4-cycles. An example illustrates wh y: Consider a vertex u whose 2-neighborho o d, the graph induced by vertices within distance 2 of u , is the co mplete bipar tite gra ph K ∆,∆ with pa rtitions X and Y . Suppose that u and ano ther vertex v are in X . If v is colo red with c in round 0 while u remains unco lored, then the set D 1 ( u, c ) = ∅ ; this vio lates equa tio n (1) since d 1 ≥ 1 if for exa mple k ≥ 2 and ∆ ≥ 2 /c 2 . So, when the gra ph has 4-cycles, d t +1 ( u, c ) is no t necessarily concentrated around its exp ectation with p ositive probability , given the state of the algo rithm at the b eginning of r ound t . W e must modify the alg orithm in t wo ways. First Mo dification: A tec hnique for colori ng graphs with 4 -cycles. While d t ( u, c ) is not co ncentrated enough when the g raph has 4-cycles, our a na lysis will sho w that the a verage o f d t +1 ( u, c ) ov er all colo r s 3 in the palette of a v ertex u is conc e ntrated enough. How do es this b enefit us? Marko v’s famous inequality may b e interpreted as: a list of s positive n umber which average d has at mo st s/q num ber s larger than q d for any p ositive n umber q . W e mo dify the algo rithm so that at the end of each round t , e very vertex u remov es from its palette ev er y colo r c with d t +1 ( u, c ) larger than 2 d t +1 . Loo k at wha t happens in round t = 1 . By a s traightforw ard application of Mar ko v’s inequality , instead of equa tio n (1) we will hav e the less stringent pr op erty: ∀ u ∈ V ( G t ) , ∀ c ∈ S t ( u ) s t ( u ) ≥ 1 2 s t (1 − o (1)) , d t ( u, c ) ≤ 2 d t (1 + o (1)) . (2) with po sitive probability . In fact, using a ge ne r alization of Ma r ko v’s inequality , the a nalysis will sho w that with a few mo re mo difica tions o ur a lgorithm maintains, with p o sitive probability , a slightly s tronger prop erty(still w eaker than eq ua tion (1)). Equation (1) implies that the s t ( u ) and d t ( u ) at all uncolored v er tices u are about the same. It is a strong statement and helps in the proofs, but is too muc h to maintain on graphs with 4-cycles. Equation (2) is w eaker and is obtained b y our algorithm with positive probability . Mor eov er , it is sufficien t to ensure that after the r epea t-until blo ck, with positive pro bability , f or eac h uncolored v er tex u a nd color c in its palette, s t ( u ) ≥ 2 d t ( u, c ). This is a key idea in our algorithm. Second M o dification: Using indep enden t random v ariables for easier analysi s. Instead o f waking up a vertex with some pr obability , and then choosing a color fr o m its palette uniformly at random; for each uncolored vertex u a nd colo r c in its palette, w e will ass ig n c to u indep endently with some prob- ability . In case multiple colors remain as s igned to the vertex after the c onflict reso lution phase, we will arbitrarily c ho o se one of them to p e rmanently color the vertex. This mo dification, adapted fro m Johansso n [1 2], will make concentration of our random v a riables simpler. Next we provide a formal description of the algorithm w e hav e j ust motiv ated. 2.2 A F ormal Description of the Alg orithm Let ( d t ) and ( s t ) be s equences defined recursively as d 0 := ∆ d t +1 := d t (1 − 1 16 e − 1 / 2 s t d t ) e − 1 / 2 s 0 := ∆/k s t +1 := s t e − 1 / 2 . (3) F or round t , vertex u , and color c , F t ( u, c ) := { c is not assigned to an y v ertex adjacen t to u in round t } (4) is an even t in the pr o bability spa ce generated b y the ra ndom c ho ices of the algorithm in ro und t , giv en the state of all data structures at the beg inning of the round. Let D esir ed F t := e − 1 / 2 . 4 Rep eat at every round t , until d t /s t < 1 / 8 Phase I - Coloring Attempt F or each v ertex u in G t , and color c in S t ( u ): Assign c to u with probability 1 4 1 d t . Phase II - Conflict R esolut ion F or each v ertex u in G t : Phase II.1 Remov e from S t ( u ), all colors assigned to adjacent vertices. Phase II.2 F or each color c in S t ( u ), remov e c from S t ( u ) with probability 1 − min (1 , D esir ed F t ( u, c ) P r ( F t ( u, c ) ) . If S t ( u ) has at least one color whic h is as s igned to u , then arbirar ily pic k an assigned color from S t ( u ) to perma nent ly color u . Phase III - Cle anup(disc ar d al l c olors c with d t +1 ( u, c ) & 2 d t +1 fr om p alette) F or each v ertex u in G t : S t +1 ( u ) = S t ( u ). Let α = 1 − | S t +1 ( u ) | /s t +1 . If α < 0, then α = 0, otherwise if α > 1 / 2, then α = 1 / 2 . Let γ b e the sma lle s t n umber in [1 , ∞ ) so that Av er ag e c ∈ S t +1 ( u ) d t +1 ( u, c ) ≤ 1 − 2 α 1 − α γ d t +1 . Remov e all colors c with d t +1 ( u, c ) ≥ 2 γ d t +1 from S t +1 ( u ). end rep eat Permanen tly color each v ertex u in V ( G t ) with a color pic ked independently and unifor mly at random from its palette S t ( u ). 3 The Main Theorem Theorem 1 (Main Theorem). Given 67 ∆/ lo g ∆ c olors and a triangle-fr e e gr aph G with m ax imu m de gr e e ∆ lar ge enough, our algorithm finds a pr op er c oloring of the gr aph with p ositive pr ob ability. W e need some lemmas to prov e the Main Theorem and befor e that w e need the following definition. Definition 3 . d t ( v ) = Av erag e c ∈ S t ( v ) d t ( v , c ) Lemma 1 (Main Lemma). Given ψ > 1 and a triangle-fr e e gr aph G with maximum de gr e e ∆ , ther e is a p ositive c onstant β such that for t he se qu enc e ( e t ) define d by e 0 = 0 , e t +1 = 3 e t + β ( r ψ s t ) for t > 0 , (5) if s t ≫ ψ and e t ≪ 1 at r ound t , then ∀ u ∈ V ( G t ) , ∃ α ∈ [0 , 1 / 2] , ∀ c ∈ S t ( u ) , 5 s t ( u ) ≥ (1 − α ) s t (1 − e t ) d t ( u ) ≤ 1 − 2 α 1 − α d t (1 + e t ) d t ( u, c ) ≤ 2 d t (1 + e t ) with p ositive pr ob ability. W e will pr ov e the Ma in Lemma in Section 5 and a ssume it in this section. Using it we can immediately conclude the following. Corollary 1 . Given the setup of the Main L emma(L emma 1), if s t ≫ ψ and e t ≪ 1 at ro und t , then ∀ u ∈ V ( G t ) , s t ( u ) ≥ 1 2 s t (1 − e t ) d t ( u ) ≤ d t (1 + e t ) with p ositive pr ob ability. Lemma 2 . The r ep e at-until blo ck finishes in 16 e 1 / 2 k r ounds. Pr o of. By the definition of sequences ( d t ) and ( s t ) in equation (3), we hav e d t +1 s t +1 = d t s t (1 − 1 16 s t d t e − 1 / 2 ) = d t s t − 1 16 e − 1 / 2 . Since d 0 s 0 = k we get d t 1 s t 1 ≤ 1 4 after 16 e 1 / 2 k r ounds. ⊓ ⊔ Let t 1 = 16 e 1 / 2 k be the last ro und of the repe at-until blo ck. Then the following lemma is a straightforw a rd applica tio n of equation (3). Lemma 3 . s t 1 = ∆ k exp ( − 8 e 1 / 2 k ) and, if k ≤ 1 9 e 1 / 2 log ∆ and ∆ is lar ge enough, then s t 1 ≫ 1 . 3.1 Bounding the Error Estimate in all Concentra tion Inequalities Now we look at s t , which is used to b ound the err or ter m e t . Lemma 4 . e t ≤ 3 t O ( r k e xp (8 e 1 / 2 k ) ψ ∆ ) . Pr o of. By Lemma 3, w e ha ve s t 1 = ∆ k exp ( − 8 e 1 / 2 k ) . Note that in equatio n (5), the r ecurrence for e t , the large st term is O ( p ψ /s t ). Since the sequence ( s t ) is decreasing, we use Lemma 3 to conclude that O ( s ψ s t 1 ) = O ( r k e xp (8 e 1 / 2 k ) ψ ∆ ) 6 is the maxim um this term can be. Thus w e can simplify the recurr ence for e t to e t +1 = 3 e t + r k e xp (8 e 1 / 2 k ) ψ ∆ ) . Since e 0 = 0 , a simple upper b ound for e t is given by e t ≤ 3 t O ( r k e xp (8 e 1 / 2 k ) ψ ∆ ) where α is some po sitive co nstant. ⊓ ⊔ Lemma 5 . Given ∆/k c olors wher e k ≤ 1 67 (log ∆ ) and a triangle-fr e e gr aph G wi th maximum de gr e e ∆ , our algorithm r e aches the end of the r ep e at-u ntil blo ck at r oun d t 1 = O ( k ) with e t 1 ≪ 1 , and ∀ u ∈ V ( G t 1 ) , c ∈ S t 1 ( u ) s t 1 ( u ) ≥ 1 2 s t 1 (1 − e t 1 ) d t 1 ( u, c ) ≤ 2 d t 1 (1 + e t 1 ) with p ositive pr ob ability. Pr o of. Let ψ = 3 log ∆ , and let t 1 be the num ber of rounds to reac h the end of th e repea t-until block. Using Lemma 4, we get e t 1 ≤ 3 t 1 O ( r k exp (8 e 1 / 2 k ) ψ ∆ ) . Using Lemma 3 it is straightf orward to show th at e t 1 ≪ 1 and s t 1 ≫ ψ if k ≤ 1 67 log ∆ . Applying Corollary 1 completes the pro of. ⊓ ⊔ W e may no w prov e the Ma in Theorem. Pr o of (of the Main The or em). Using Lemma 5, we g et ∀ u ∈ V ( G t 1 ) , c ∈ S t 1 ( u ) s t 1 ( u ) ≥ 2 d t 1 ( u, c ) with p ositive probability . No w Haxell [11] s hows tha t the final step of our a lgorithm finds(ra ndo mly coloring all uncolored vertices) finds a pr op er coloring with positive probabilit y . ⊓ ⊔ 4 Sev eral Useful Inequalities Now we lo o k at some preliminaries which will be used in the pr o of details. The next lemma describ es what happ ens to the av era g e v alue of a finite subset of real n umber s when large elemen ts ar e remov ed. As s hown in the statement of the lemma, it implies Markov’s Inequality [4]. Lemma 6 . Consider a set of p ositive r e al numb ers of size n and a ver age value µ . If we r emove αn elements with value atle ast q µ for some q > 1 , then t he r emaining p oints have aver age µ ′ ≤ µ 1 − q α 1 − α . In p articular, α ≤ 1 q sinc e µ ′ ≥ 0 . 7 Pr o of. The conclusio n is obta ine d by a triv ial manipulation of the fo llowing ineq uality whic h relates µ and µ ′ . q µα + µ ′ (1 − α ) ≤ µ ⊓ ⊔ The next lemma describ es what happ ens when w e add large elemen ts to a finite subset of real num b ers. Lemma 7 . Given the setup o f L emma 6 , if we add αn p oints with value qµ to the sample, then the r esu lt ing lar ger sample has aver age µ ′ = µ 1 + q α 1 + α Pr o of. The conclusion is easily obtained from the follo wing equation relating µ and µ ′ . µ ′ (1 + α ) = µ + q µα ⊓ ⊔ W e use the following le mma for computations with error factors. Lemma 8 . L et ( A n ) b e a se quenc e such t hat 0 < A n < c < 1 (wher e c is a c onstant) , and let ( e n ) b e a nother se quenc e. Then 1 − A n (1 + e n ) = (1 − A n )(1 + O ( e n )) Pr o of. 1 − A n (1 + e n ) = (1 − A n )(1 + e n ) − e n = (1 − A n )(1 + e n ) − (1 − A n ) e n (1 − A n ) = (1 − A n )(1 + e n ) − (1 − A n ) O ( e n ) = (1 − A n )(1 + O ( e n )) ⊓ ⊔ W e use the following version of Azuma’s inequality [20] to prov e concen tra tion of random v a riables. Theorem A (Azuma’s inequality ) L et X b e a r andom variable determine d by n trials T 1 , . . . , T n , such that for e ach i , and any two p ossible se quenc es of outc omes t 1 , . . . , t i and t 1 , . . . , t i − 1 , t ′ i : | E [ X | T 1 = t 1 , . . . , T i = t i ] − E [ X | T 1 = t 1 , . . . , T i = t ′ i ] | ≤ α i then P r ( | X − E [ X ] | > t ) ≤ 2 e − t 2 / ( P α 2 i ) W e use the following version of the Lo v asz Local Lemma [20] Theorem B (Lo v asz Lo cal Lemma) Consider a set E of events such that for e ach A ∈ E • P r ( A ) ≤ p < 1 , and • A is mutual ly indep endent of a set of al l but at most d of the other event s. If 4 pd ≤ 1 , then with p ositive pr ob ability, none of the events in E o c cur. 8 5 Pro of of the Main Lemma The following as s umptions are repe atedly used in the lemmas of this section. Assumption 1 Assu me s t ≫ ψ, e t ≪ 1 and with p ositive pr ob ability ∀ u ∈ V ( G t ) , ∀ c ∈ S t ( u ) , ∃ α ∈ [0 , 1 / 2] , s t ( u ) ≥ (1 − α ) s t (1 − e t ) d t ( u ) ≤ 1 − 2 α 1 − α d t (1 + e t ) d t ( u, c ) ≤ 2 d t (1 + e t ) . All events in this section are in the probability space generated by our randomized algor ithm in r ound t + 1, given the sta te of all data structures at the beginning of the round. Pr o of (of the Main L emm a). The pro of is by induction on the round num b er t us ing lemma s that fo llow. The base case, when t = 0, is trivially true. If w e assume Assumption 1, the induction h y po thesis, for round t then for each vertex u in V ( G t +1 ), b y Lemma 13 w e have P r { d t +1 ( u ) ≤ d t +1 (1 + O ( e t + r ψ s t + 1 d t ) } ≥ 1 − e − ψ O (1) . F or each c in S t +1 ( u ), by Lemma 1 4, w e ha ve P r {∃ α ∈ [0 , 1 2 ] such that s t +1 ( u ) ≥ (1 − α ) s t +1 (1 − 3 e t + O ( r ψ s t + 1 d t )) , d t +1 ( u ) ≤ 1 − 2 α 1 − α d t +1 (1 + 3 e t + O ( r ψ s t + 1 d t )) , d t +1 ( u, c ) ≥ (1 − α )2 d t +1 (1 − 3 e t + O ( r ψ s t + 1 d t )) } ≥ 1 − e − ψ O (1) . Each of the even ts in the probabilities a bove is dependent on at most O ( ∆ 2 ) other such even ts. If ψ ≥ 3 log ∆ and ∆ is large enough, then w e use Theor em B to co nclude that Assumption 1 holds for round t + 1 . ⊓ ⊔ The ab ove proo f of the Main Lemma require d L e mma s 13 and 14. The rest of this section will prov e thes e lemmas. Next w e consider the state of the palettes just befor e the clean up phase of round t . Definition 4 . L et ˜ S t ( u ) b e t he list of c olors in the p alette of vertex u in r ound t just b efor e t he cle anup phase, and let ˜ s t ( u ) b e the size of ˜ S t ( u ) . That is, ˜ S t ( u ) is obtaine d fr om S t ( u ) by r emoving c olors disc ar de d in t he c onfl ict r esolution phase. Lemma 9 . Given Assu mption 1, for e ach vertex u in V ( G t +1 ) we have P r { s t ( u ) e − 1 / 2 (1 − 1 2 e t − O ( r ψ s t )) ≤ ˜ s t ( u ) ≤ s t ( u ) e − 1 / 2 (1 + O ( r ψ s t )) } ≥ 1 − e − ψ O (1) . 9 Pr o of. Suppos e u is an uncolored v ertex at the beginning o f round t , and c a co lor in its palette. P r { c is removed from S t ( u ) in phase II.1 } = 1 − P r { no neighbor of u is assigned c } = 1 − Y v ∈ D t ( u,c ) (1 − P r { v is assig ned c } ) = 1 − Y v ∈ D u,c (1 − 1 4 1 d t ) ≤ 1 − (1 − 1 4 1 d t ) d t ( u,c ) ≤ 1 − (1 − 1 4 1 d t ) 2 d t (1+ e t ) ≤ 1 − e log(1 − 1 4 1 d t )2 d t (1+ e t ) h log(1 + x ) = x + O ( x 2 ) i ≤ 1 − e ( − 1 4 1 d t + O ( 1 d t ) 2 )2 d t (1+ e t ) h Assumption 1 i ≤ 1 − e − 1 / 2 (1 − 1 2 e t + O ( 1 d t )) In phase II.2 of round t + 1 w e remo ve colors fro m the palette using an appropriate b erno ulli v aria ble, to get P r { c / ∈ ˜ S t ( u ) } = 1 − e − 1 / 2 (1 − 1 2 e t + O ( 1 d t )) . Using linearity o f expec tation E [ ˜ s t ( u )] = s t ( u ) e − 1 / 2 (1 − 1 2 e t + O ( 1 d t )) . F or concentration of ˜ s t ( u ), suppose s t ( u ) = m . Let c 1 , . . . , c m be the colo r s in S t ( u ). Then ˜ S t ( u ) may b e considered a rando m v a riable determined by m trials T 1 , . . . , T m where T i is the set of vertices in G t that are a ssigned color c i in r o und t . Observe that T i affects ˜ S t ( u ) by at most 1 given T 1 , . . . , T i − 1 . Now us ing Theorem A we get, P r {| ˜ s t ( u ) − E [˜ s t ( u )] | ≥ p ψ s t ( u ) } ≤ e − ψ O (1) . ⊓ ⊔ W e now f o cus on the sets D t ( u, c ). The following t wo lemmas will help. Lemma 1 0. L et u b e an unc olor e d vertex , and c b e a c olor in its p alette at the b e ginning of r ound t . Then given Assu mption 1, we have P r { u is assigne d c and c ∈ ˜ S t ( u ) } = 1 4 1 d t e − 1 / 2 (1 − 1 2 e t + O ( 1 d t )) . Pr o of. P r { u is ass ig ned c and c ∈ ˜ S t ( u ) } = P r { u is assig ned c } P r { c ∈ ˜ S t ( u ) } = 1 4 1 d t e − 1 / 2 (1 − 1 2 e t + O ( 1 d t )) h Equation ( ?? ) i ⊓ ⊔ The following lemma is a consequence of the prev io us one. 10 Lemma 1 1. L et u b e an unc olor e d vertex at t he b e ginn ing of r oun d t . Then given Assu mption 1, we have P r { u is c olor e d } ≥ 1 16 s t d t e − 1 / 2 (1 − 3 e t + O ( 1 d t )) . Pr o of. Consider the even t { u is colored } = [ c ∈ S t ( u ) { u is assigned c and c ∈ ˜ S t ( u ) } . Since the even ts in the union on the righ t hand side of the equation above are indep endent , P r { u is colo red } = 1 − Y c ∈ S t ( u ) (1 − P r { u is assig ned c and c ∈ ˜ S t ( u ) } ) . Now using Lemma 10, we get P r { u is colored } ≥ 1 − (1 − 1 4 1 d t e − 1 / 2 (1 − 1 2 e t + O ( 1 d t ))) s t ( u ) ≥ 1 − (1 − 1 4 1 d t e − 1 / 2 (1 − 1 2 e t + O ( 1 d t ))) 1 2 s t (1 − e t ) h Assumption 1 i ≥ 1 − e xp ( − 1 8 s t d t e − 1 / 2 (1 − 3 2 e t + O ( 1 d t ))) ≥ 1 − (1 − 1 16 s t d t e − 1 / 2 (1 − 3 2 e t + O ( 1 d t ))) = 1 16 s t d t e − 1 / 2 (1 − 3 2 e t + O ( 1 d t )) . ⊓ ⊔ W e will need the following definitio ns . Definition 5 . • L et ˜ D t ( u, c ) b e the set of unc olor e d vertic es that have c olor c in their p alettes and ar e un c olor e d in r ound t , just b efor e the cle anup phase. That is, ˜ D t ( u, c ) = D t ( u, c ) \ ( { v | c / ∈ ˜ S t ( v ) } ∪ { v | v is c olor e d in r oun d t } ) . • L et ˜ d t ( u, c ) b e the s ize of ˜ D t ( u, c ) . • ¯ d t ( u ) := P c ∈ ˜ S t ( u ) ˜ d t ( u, c ) = P c ∈ S t ( u ) 1 { c ∈ ˜ S t ( u ) } ˜ d t ( u, c ) • ˜ d t ( u ) := ¯ d t ( u ) ˜ s t ( u ) Lemma 1 2. Given Assumption 1, for e ach vertex u in V ( G t +1 ) we have P r { ˜ d t ( u ) ≤ d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t + s ψ d t s t d t ( u ) )) } ≥ 1 − e − ψ O (1) . 11 Pr o of. Let u b e a n uncolore d vertex at the b eginning of round t , a nd let c b e a color in its pa lette. F o r a vertex v in D t ( u, c ), Lemma 10 implies that P r { v is c o lored with d } = O (1 / d t ) for an y color d in S t ( v ). Thu s, P r ( { c / ∈ ˜ S t ( v ) } ∩ { v is color e d } ) = X d ∈ S t ( v ) P r ( { c / ∈ ˜ S t ( v ) } ∩ { v is color e d with d } ) = X d ∈ S t ( v ) P r { c / ∈ ˜ S t ( v ) | v is color ed with d } P r { v is colo red with d } = P r { c / ∈ ˜ S t ( v ) } (1 + O ( 1 d t )) X d ∈ S t ( v ) P r { v is color ed with d } = P r { c / ∈ ˜ S t ( v ) } P r { v is colored } (1 + O ( 1 d t )) . A straightforw ard computation now shows that P r ( { c / ∈ ˜ S t ( v ) } ∩ { v is not color ed } ) = P r { c / ∈ ˜ S t ( v ) } P r { v is not colored } (1 + O ( 1 d t )) . (6) Now, v is r emov ed fr om the set D t ( u, c ) if e ither it is colored o r colo r c is removed fro m its palette. This means that even t { v / ∈ ˜ D t ( u, c ) } = { v is colored } ∪ ( { c / ∈ ˜ S t ( v ) } ∩ { v is not color ed } ) . Since G is triangle-free, u and v do not hav e an y common neigh bo r s. This implies that P r { v / ∈ ˜ D t ( u, c ) | c ∈ ˜ S t ( u ) } = P r { v / ∈ ˜ D t ( u, c ) } (1 + O ( 1 d t )) = ( P r { v is colo red } + P r ( { c / ∈ ˜ S t ( v ) } ∩ { v is not co lored } ))(1 + O ( 1 d t )) = ( P r { v is colo red } + P r { c / ∈ ˜ S t ( v ) } P r { v is not color e d } )(1 + O ( 1 d t )) h equation (6) i = ( P r { v is colo red } + (1 − e − 1 / 2 )(1 − P r { v is co lored } ))(1 + O ( 1 d t )) h equation ( ?? ) i = (1 − (1 − P r { v is colo r ed } ) e − 1 / 2 )(1 + O ( 1 d t )) ≥ (1 − (1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 )(1 + 2 e t + O ( 1 d t )) h Lemma 11 i . Using linearity o f expec tation E [ ˜ d t ( u, c ) | c ∈ ˜ S t ( u )] = E [ ˜ d t ( u, c )](1 + O ( 1 d t )) ≤ d t ( u, c )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( 1 d t )) . (7) Now using the ab ov e bound E [ ¯ d t ( u )] = X c ∈ S t ( u ) P r { c ∈ ˜ S t ( u ) } E [ ˜ d t ( u, c ) | c ∈ ˜ S t ( u )] ≤ e − 1 / 2 X c ∈ S t ( u ) d t ( u, c )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( 1 d t )) ≤ e − 1 / 2 s t ( u ) d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( 1 d t )) 12 F or concentration of ¯ d t ( u ), suppose s t ( u ) = m . Let c 1 , . . . , c m be the colors in S t ( u ). Then ¯ d t ( u ) ma y be considered a random v ariable determined by the random tria ls T 1 , . . . , T m , where T i is the set of vertices in G t that are assigned color c i in round t . Observe that T i affects ¯ d t ( u ) b y a t most d t ( u, c ). Thu s P α 2 i in the s tatement of Theorem A is less th at P c ∈ S t ( u ) d 2 t ( u, c ). This upperb ound is maximized when the d t ( u, c ) take the extreme v alues of 2 d t and 0 sub ject to d t ( u ) = 1 s t ( u ) P c ∈ S t ( u ) d t ( u, c ). Thus X α 2 i ≤ O (( d t ) 2 d t ( u ) s t ( u ) /d t ) ≤ O ( s t ( u ) d t d t ( u )) Using Theorem A, w e get P r { ¯ d t ( u ) − e − 1 / 2 s t ( u ) d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( 1 d t )) ≥ O ( p ψ s t ( u ) d t d t ( u )) } ≤ e − ψ O (1) . Lemma 9 says that P r { s t ( u ) e − 1 / 2 (1 − 1 2 + O ( r ψ s t )) ≤ ˜ s t ( u ) ≤ s t ( u ) e − 1 / 2 (1 + O ( r ψ s t )) } ≥ 1 − e − ψ O (1) . Combining the a b ove tw o inequalities we ha ve P r { ¯ d t ( u ) ˜ s t ( u ) − d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( 1 d t + r ψ s t )) ≥ O ( s ψ d t d t ( u ) s t ) } ≤ e − ψ O (1) . Therefore P r { ¯ d t ( u ) ˜ s t ( u ) ≥ d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t + s ψ d t s t d t ( u ) )) } ≤ e − ψ O (1) . ⊓ ⊔ Note that ¯ d t ( u ) ˜ s t ( u ) is the average | ˜ D t ( u, c ) | at a vertex u at the end phase II. Phas e I I I only br ings this av era ge down b y removing colors with large d u,c . Thus we get the next lemma almost immediately . Lemma 1 3. Given Assumption 1, for e ach u in V ( G t +1 ) we have P r { d t +1 ( u ) ≤ d t +1 (1 + 2 e t + O ( r ψ s t + 1 d t ) } ≥ 1 − e − ψ O (1) . Pr o of. Let u be a vertex in V ( G t +1 ). By Lemma 12 P r { ˜ d t ( u ) ≤ d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t + s ψ d t s t d t ( u ) )) } ≥ 1 − e − ψ O (1) . 13 Now d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t + s ψ d t s t d t ( u ) )) = d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t )) + O ( s ψ d t ( u ) d t s t ) = d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t )) + d t O ( s ψ d t ( u ) s t d t ) ≤ d t (1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t )) + d t O ( r ψ s t ) = d t (1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t )) . Thu s the ev ent { ¯ d t ( u ) ˜ s t ( u ) ≥ d t (1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t )) } ⊆ { ¯ d t ( u ) ˜ s t ( u ) ≥ d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t + s ψ d t s t d t ( u ) )) } . Therefore e − ψ O (1) ≤ P r { ¯ d t ( u ) ˜ s t ( u ) ≥ d t (1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t )) } ≤ P r { ¯ d t ( u ) ˜ s t ( u ) ≥ d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 2 e t + O ( r ψ s t + 1 d t + s ψ d t s t d t ( u ) )) } . ⊓ ⊔ Next we sho w that in the clea nup phase o f round t , a vertex disca rds so many colors th at its pa le tte size in round t + 1 bec o mes less than 1 2 s t +1 (1 − e t +1 ) with a v ery small probability . Lemma 1 4. Given Assumption 1, for e ach vertex u in V ( G t +1 ) we have P r {∃ α ∈ [0 , 1 2 ] s uch that ∀ c ∈ S t +1 ( u ) s t +1 ( u ) ≥ (1 − α ) s t +1 (1 − 3 2 e t + O ( r ψ s t + 1 d t )) , d t +1 ( u ) ≤ 1 − 2 α 1 − α d t +1 (1 + 3 2 e t + O ( r ψ s t + 1 d t )) , d t +1 ( u, c ) ≤ 2 d t +1 (1 + 3 2 e t + O ( r ψ s t + 1 d t )) } ≥ 1 − e − ψ O (1) . Pr o of. Consider vertex u ∈ V ( G t +1 ). Using Assumption 1 , at ro und t , ∃ α ∈ [0 , 1 2 ] such that s t ( u ) ≥ (1 − α ) s t (1 − e t ) and d t ( u ) ≤ 1 − 2 α 1 − α d t (1 + e t ). By Lemma 9 w e get P r { s t ( u ) e − 1 / 2 (1 − 1 2 e t + O ( r ψ s t )) ≤ ˜ s t ( u ) ≤ s t ( u ) e − 1 / 2 (1 + 1 2 e t + O ( r ψ s t )) } ≥ 1 − e − ψ O (1) . 14 Now ˜ s t ( u ) = s t ( u ) e − 1 / 2 (1 + 1 2 e t + O ( r ψ s t + 1 d t )) ≥ (1 − α ) s t e − 1 / 2 (1 + 1 2 e t + O ( r ψ s t + 1 d t )) ≥ (1 − α ) s t +1 (1 + 1 2 e t + O ( r ψ s t + 1 d t )) . By Lemma 12 w e g et P r { ˜ d t ( u ) ≤ d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 3 2 e t + O ( r ψ s t + 1 d t + s ψ d t s t d t ( u ) )) } ≥ 1 − e − ψ O (1) . Now ˜ d t ( u ) ≤ d t ( u )(1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 3 2 e t + O ( r ψ s t + 1 d t )) ≤ 1 − 2 α 1 − α d t (1 − 1 16 s t d t e − 1 / 2 ) e − 1 / 2 (1 + 3 2 e t + O ( r ψ s t + 1 d t )) ≤ 1 − 2 α 1 − α γ d t +1 . where γ is the smallest num b er in [1 , ∞ ) for which the ab ove inequality is true. Combining the pr eceding inequalities, we get γ = 1 + 3 e t + O ( r ψ s t + 1 d t ) . In the cleanup phase o f our algor ithm(given in Section 2 .2), the change in pa lette is equiv alent to the following process. 1. Add α 1 − α ˜ s t ( u ) arbitrary colors to u ’s pa lette, w ith ˜ d t ( u, c ) = 2 γ d t +1 . This adjusts the palette size to ˜ s t ( u ) ≥ s t +1 (1 + 3 e t + O ( p ψ /s t + 1 /d t )). L e mma 7 ensur e s that the adjusted new average is ˜ d t ( u ) ≤ γ d t +1 2. Remo ve all the colors with d t ( u, c ) ≥ 2 γ d t +1 . Now we use Lemma 6, s etting µ to γ d t +1 and q µ to 2 γ d t +1 , to get s t +1 ( u ) ≥ (1 − α ) s t +1 (1 + 3 e t + O ( r ψ s t + 1 d t )) and d t +1 ( u ) ≤ 1 − 2 α 1 − α d t +1 (1 + 3 e t + O ( r ψ s t + 1 d t )) . The result is obtained using Lemmas 9 and 12. ⊓ ⊔ 6 Ac kno wledgemen t I am indebted to F an Ch ung her comments and supp ort. 15 References 1. Dimitris Achlioptas and b o oktitle = FOCS year = 1997 pages = 204-212 b ib source = DBLP , http://dblp.uni- trier.de Michael Molloy , titl e = The anal ysis of a list-coloring algorithm on a random graph. 2. Mikl´ os Ajtai, J´ anos K oml´ os, and Endre Szemer ´ edi, A dense infinite Sidon se quenc e , Europ ean J. Com b in. 2 (1981), n o. 1, 1–1 1. MR 611925 (83f: 10056) 3. Noga Alon, Michael Krivelevic h , and Benny Sudako v , Coloring gr aphs wi th sp arse nei ghb orho o ds , J. Combin. Theory Ser. B 77 (199 9), no. 1, 73– 82. MR 1710532 (2001a :05054) 4. Noga Alon and Joel H. Spencer, The pr ob abilistic metho d , third ed., Wil ey-Interscience Series in Discrete Math- ematics and Optimization, John Wile y & Sons In c., Hob oken, NJ, 2008, With an app end ix on the life and w ork of Paul Erd˝ os. M R 2437651 (2009j:60 004) 5. B ´ ela Bollob´ as, Chr omatic numb er, girth and maximal de gr e e , Discrete Math. 24 (1978), no. 3, 311–314 . MR 523321 (80 e:05058 ) 6. O. V. Boro din and A . V. Kosto chk a, On an upp er b ound of a gr aph’s chr omatic numb er, dep ending on the gr aph’s de gr e e and density , J. Combinatorial Theory Ser. B 23 (1977), n o. 2-3, 247– 250. MR 0 469803 (57 #9584) 7. R. L. Bro oks, On c olouring the no des of a network , Proc. Cambridge Philos. S oc. 37 (1941), 194–197. MR 0012236 (6,281b) 8. P aul A. Catlin, A b ound on t he chr omatic numb er of a gr aph , Discrete Math. 22 (1978), no. 1, 81–8 3. MR 522914 (80a:050 90a) 9. P . F rankl and V. R¨ odl, Ne ar p erfe ct c overings i n gr aphs and hyp er gr aphs , Europ ean J. C om bin. 6 (19 85), no. 4 , 317–326 . MR 829351 (88a:0511 6) 10. Mic hael R. Garey and David S. Johnson, Computers and intr actability: A g uide t o the t he ory of np-c ompl eteness , W.H. F reeman, New Y ork, 1 979. 11. P . E. Haxell, A note on vertex list c olouring , Com bin. Probab. Comput. 10 (2001), no. 4, 345–347 . MR 1860440 (2002g:0 5082) 12. A. Johansson, Asympto tic choic e numb er for triangle fr e e gr aphs , Unpublished, see Mol lo y and R eed [20]. 13. Jeff Kahn, Coloring ne arly-disjoint hyp er gr aphs with n + o ( n ) c olors , J. Combin. Theory Ser. A 59 (1992), no. 1, 31–39. M R 1141320 (93b:0512 7) 14. , Asymptotic al ly go o d list-c ol orings , J. Combin. Theory Ser. A 73 (1996), n o. 1, 1–59. MR 1367606 (96j:050 01) 15. Subhash Khot, Impr ove d inapr oximabili ty r esults for maxclique, chr omatic numb er and appr oximate gr aph c olor- ing , FOCS, 2001, pp. 600–609. 16. Jeong H an K im, On Br o oks’ the or em for sp arse gr aphs , Com bin. Probab. Comput. 4 (1995), no. 2, 97– 132. MR 1342856 (96f:05078) 17. , The Ramsey numb er R (3 , t ) has or der of magnitude t 2 / log t , Random Structures Algorithms 7 ( 1995), no. 3, 173–2 07. MR 1369063 (96m:05140 ) 18. A. V . Kostoˇ ck a and N. P . Masurov a, An estimate in the the ory of gr aph c oloring , Diskret. Analiz (1977), no. 30 Metody Diskret. An al. v Resenii Kombinatorn y h Zadac, 23–29, 76. MR 05438 05 (58 #27604) 19. Jim Lawrence, Covering the vertex set of a gr aph with sub gr aphs of smal l er de gr e e , Discrete Math. 21 (1978), no. 1, 61–68 . MR 523419 (80a: 05094) 20. Mic hael Mollo y and Bruce Reed, Gr aph c olouring and the pr ob abilistic metho d , Algorithms and Combinatorics, vol . 23, Springer-V erlag, Berl in, 2002 . MR 1869439 (2003c:0 5001) 21. Nic holas Pipp enger and Joel S p encer, Asymptotic b ehavior of the chr omatic i ndex for hyp er gr aphs , J. Com bin. Theory Ser. A 51 (19 89), no. 1, 24– 42. MR 993646 (90h:0 5091) 22. V. G. Vizing, Some unsolve d pr oblems in gr aph the ory , Usp ehi Mat. Nauk 23 (1968), no. 6 (144), 117–134. MR 0240000 (39 #1354) 23. V an H. V u , A gener al upp er b ound on the li st chr omatic numb er of lo c al ly sp arse gr aphs , Com b in . Probab. Comput. 11 (2002), no. 1, 103 –111. MR 1888186 (2003c:0 5090) 24. Nic holas C. W ormald, Di ffer ential e quations f or r andom pr o c esses and r andom gr aphs , Ann. A p pl. Probab. 5 (1995), n o. 4, 121 7–1235. MR 1384372 (97c:0513 9) 16