On the game chromatic number of sparse random graphs

Transcription

1 O the game chromatic umber of sparse radom graphs Ala Frieze ad Simcha Haber ad Mikhail Lavrov Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh PA, 15213, USA. Jue 8, 2018 Abstract Give a graph G ad a iteger k, two players take turs colorig the vertices of G oe by oe usig k colors so that eighborig vertices get differet colors. The first player wis iff at the ed of the game all the vertices of G are colored. The game chromatic umber χ g G is the miimum k for which the first player has a wiig strategy. The paper [6] bega the aalysis of the asymptotic behavior of this parameter for a radom graph G,p. This paper provides some further aalysis for graphs with costat average degree i.e. p = O1 ad for radom regular graphs. We show that w.h.p. c 1 χg,p χ g G,p c 2 χg,p for some absolute costats 1 < c 1 < c 2. We also prove that if G,3 deotes a radom -vertex cubic graph the w.h.p. χ g G,3 = 4. 1 Itroductio Let G = V, E be a graph ad let k be a positive iteger. Cosider the followig game i which two players Alice ad Bob take turs i colorig the vertices of G with k colors. Each move cosists of choosig a ucolored vertex of the graph ad assigig to it a color from {1,..., k} so that the resultig colorig is proper, i.e., adjacet vertices get differet colors. A wis if all the vertices of G are evetually colored. B wis if at some poit i the game the curret partial colorig caot be exteded to a complete colorig of G, i.e., there is a ucolored vertex such that each of the k colors appears at least oce i its eighborhood. Research supported i part by NSF Grat CCF ala@radom.math.cmu.edu. simi@adrew.cmu.edu. mlavrov@adrew.cmu.edu. 1

2 We assume that A goes first our results will ot be sesitive to this choice. The game chromatic umber χ g G is the least iteger k for which A has a wiig strategy. This parameter is well defied, sice it is easy to see that A always wis if the umber of colors is larger tha the maximum degree of G. Clearly, χ g G is at least as large as the ordiary chromatic umber χg, but it ca be cosiderably more. The game was first cosidered by Brams about 25 years ago i the cotext of colorig plaar graphs ad was described i Marti Garder s colum [12] i Scietific America i The game remaied uoticed by the graph-theoretic commuity util Bodlaeder [5] re-iveted it. For a survey see Barticki, Grytczuk, Kierstead ad Zhu [4]. I this paper, we study the game chromatic umber of the radom graph G,p ad the radom d-regular graph G,d. Defie b = 1. The followig estimates were proved i 1 p Bohma, Frieze ad Sudakov [6]. Theorem 1.1. a There exists K > 0 such that for ε > 0 ad p l Kε 3 / we have that w.h.p. 1 χ g G,p 1 ε log b p. b If α > 2 is a costat, K = max{ 2α α 1, α α 2 } ad p l K / the w.h.p. χ g G,p α log b p. I this paper we complemet these results by cosiderig the case where p = d where d is at least some sufficietly large costat. We will assume that d 1/4 sice Theorem 1.1 covers larger d. Theorem 1.2. Let p = d where d is larger tha some absolute costat ad d 1/4. a If α < 4 7 is a costat the w.h.p. χ g G,p αd l d. b If α is a sufficietly large costat the w.h.p. χ g G,p αd l d. Note that whe p = o1 we have d. Note also that the bouds i Theorem 1.1 log b p l d are stroger tha those i Theorem 1.2, wheever both results are applicable. 1 A sequece of evets E occurs with high probability w.h.p. if lim PE = 1 2

3 It is atural to compare our bouds with the asymptotic behavior of the ordiary chromatic umber of radom graph. It is kow by the results of Bollobás [7] ad Luczak [16] that whe p = o1, χg,p = 1+o1 d w.h.p. Of course a stroger result is ow kow, see 2 l d Achlioptas ad Naor [2]. Thus Theorem 1.2 shows that the game chromatic umber of G,p is at most roughly twelve times ad at least roughly 8/7 times its chromatic umber. Havig proved Theorem 1.2, we exted the results to the radom d-regular graph G,d. Theorem 1.3. Let ε > 0 be a arbitrary costat. a If α is a costat satisfyig the coditios of Theorem 1.1 or Theorem 1.2 where appropriate ad d is sufficietly large ad d 1/3 ε the w.h.p. χ g G,d αd l d. b If α is a costat satisfyig the coditios of Theorem 1.1 or Theorem 1.2 where appropriate ad d is sufficietly large ad d 1/3 ε the w.h.p. χ g G,d αd l d. It is kow by the result of Frieze ad Luczak [11] that w.h.p. χg,d = 1 + o1 d 2 l d. Of course stroger results are ow kow, see Achlioptas ad Moore [1] ad Kemkes, Péres- Giméez ad Wormald [14]. Theorem 1.3 says othig about χ g G,d whe d is small. We have bee able to prove Theorem 1.4. If d = 3 the w.h.p. χg,d = 4. It is easy to see via Brooks theorem that w.h.p. the chromatic umber of a radom cubic graph is three ad so Theorem 1.4 separates χ ad χ g i this cotext. We ofte refer to the followig Cheroff-type bouds for the tails of biomial distributios see, e.g., [3] or [13]. Let X = i=1 X i be a sum of idepedet idicator radom variables such that PX i = 1 = p i ad let p = p p /. The PX 1 εp e ε2 p/2, 1.1 PX 1 + εp e ε2 p/3, ε 1, 1.2 PX µp e/µ µp Outlie of the paper Sectio 2 is devoted to the proof of Theorem 1.2. I Sectio 2.1, we prove a lower boud o χ g G,p by givig a strategy for player B. Basically, B s strategy is to follow A colorig a vertex with color i by colorig a radom vertex v with color i. Of course we mea here 3

4 that v is radomly chose from vertices outside of the eighborhood of the set of vertices of color i. Why does this work? Well, it is kow that choosig a idepedet set via a greedy algorithm will w.h.p. fid a idepedet set that is about oe half the size of the largest idepedet set. What we show is that choosig radomly half the time also has a deleterious effect o the size of the idepedet set color class selected. This leads to the game chromatic umber beig sigificatly larger tha the chromatic umber. I Sectio 2.2, we prove a upper boud o χ g G,p by givig a strategy for player A. Here A follows the same strategy used i the proof of Theorem 1.1b, up util close to the ed. We the let A follow a more sophisticated strategy. A s iitial strategy is to choose a vertex with as few available colors ad color it with ay available color i.e. oe that does ot coflict with its colored eighbors. At a certai poit there are few ucolored vertices ad they all have a substatial umber of available colors. We show that the edges of the graph iduced by these vertices ca be partitioed ito a forest F plus a low degree subgraph. Usig the tree colorig strategy described i [10] we see that the low degree subgraph does ot prevet G from beig colored. Havig proved Theorem 1.2 we trasfer the results to radom d-regular graphs d 1/4 by showig that the uderlyig structural lemmas remai true or trivially modified. This is doe i Sectio 3. I Sectio 3 we show how to covert Theorem 1.1 ito a radom regular graph settig. The two rages d 0 d 1/4 ad 1/4 < d 1/3 ε are treated seperately. The lower rage is treated i Sectio 3.1 ad the upper rage is treated i Sectio 3.2 usig the Sadwichig Theorem of Kim ad Vu [15]. I Sectio 4 we provide a strategy for B showig that w.h.p. χ g G,3 = 4. This proves Theorem 1.4. B s strategy is based o his ability to force A ito playig o a small set of vertices. B will the make a sequece of such forcig moves alog a cycle to create a double threat ad wi the game. 2 Theorem 1.2: G,p, p = d/ 2.1 The lower boud Let D = d ad suppose that there are k = αd colors. At ay stage, let C l d i be the set of vertices that have bee colored i ad let C = k i=1 C i. Let U = [] \ C be the set of ucolored vertices ad let U i = U \ NC i. Note that [] = {1, 2,..., } is the vertex set of G,p. B s strategy will be to choose the same color i that A just chose ad the to assig color i to a radom vertex i U i. The idea beig that makig radom choices whe costructig a idepedet set color class teds to oly get oe of half the maximum size. A could be makig better choices ad so we do ot maage to prove that we eed twice as may colors 4

5 as the chromatic umber. Suppose that we ru this process for θ rouds ad that C i = 2β/D where we will later take θ = 7α/8 < 1/2 ad β = 1/2. Let S i be the set of β/d vertices i C i that were colored by B. We cosider the probability that there exists a set T of size γ/d such that C i T is idepedet. For expressios X, Y we use the otatio X b Y i place of X = OY whe the bracketig is ugly. P C i, T b k PS i = S1 p 2β+γ2 2 /2D β/d γ/d S =β/d k β/d! β/d γ/d S =β/d β/d j=1 7 2 /2D2 1 1 p 2j p2β+γ2 1 2θ 2 β/d! 7 β/d 2β+γ22/2D2 k 1 p β/d γ/d 1 2θ β/d 1 p β2 2 /D 2 ed β β γ 7 ed k exp { 2β 2 2β + γ 2 d/2d } /D β 1 2θ γ 2.2 = k exp { β + γ + β 2 2β + γ 2 /2 + o d 1d 1 l 2 d } 2.3 = o1 if 2β + γ 2 > 2β + β 2 + γ. This is satisfied whe β = 1/2 ad γ = 3/4. We will justify 2.1 ad 2.2 mometarily. If the evet { C i, T } does ot occur the because o color class has size greater tha 2β + γ/d the umber l of colors i for which S i β/d by this time satisfies l2β + γ D + 2k lβ D We choose θ = 7α/8. Sice k l, this implies that 2θ. k D 2θ 2β + γ = α. This completes the proof of Part a of Theorem 1.2. Justifyig 2.1: Here we are takig the uio boud over all β/d γ/d possible choices of C i \S i ad T. I some sese we are allowig player A to simultaeously choose all possible sets of size β/d for C i \ S i. The uio boud shows that w.h.p. all choices fail. We do ot sum over orderigs of C i \ S i. We istead compute a upper boud o PS i = S that holds regardless of the order i which A plays. We cosider the situatio after θ rouds. That is, we thik of the followig radom process: pick a graph G G, p, let Alice play the colorig game o G with k colors agaist a player who radomly chooses a available vertex to be colored by the same color as Alice. Stop after θ moves. At this poit Alice played 5

6 with color i ad there are β/d vertices that were colored i by Alice ad the same umber that were colored i by Bob. We boud the probability that at this poit there are γ/d vertices that form a idepedet set with the i th color class. We take a uio boud over all the possible sets for Alice s vertices ad for the vertices i T. The probability of Bob choosig a certai set is computed below. Justifyig 2.2: For this we first cosider a sequece of radom variables X 1 = N = 1 2θ, X j = BiX j 1, q where q = 1 p 2 ad 1 j t. X j is a lower boud for the umber of vertices Bob ca color i. The probability that a vertex was i-available at time j 1 ad is still i-available ow is 1 p 2. This is because two more vertices have bee colored i. Also, we take X 1 = N as a lower boud o the umber of choices at the start of the process. The we estimate EY t where { 0 Xt = 0 Y t = 1 X 1 X 2 X t X t > 0 We use Y βd as a upper boud o the probability that B s sequece of choices is x 1, x 2,..., x β/d where S = { x 1, x 2,..., x β/d }. The term Xj lower bouds the umber of choices that B has ad so 1/X j upper bouds the probability that B chooses x j. We take the expectatio of the product of these bouds over G,p. Now if B = Biν, q ad we take k i=1 k 1 ν k E = B + i 1 i=1 l=1 i=1 = 1 q k k i=1 1 q k 1 B+i 1 1 l + i 1 1 ν + i k i=1 Suppose ow that q = 1 o1. The ν l=1 k l ν + k q l+k 1 q ν l l + k k/2 l=1 1 ν + i k l = 0 whe B = 0 the ν q l 1 q ν l l ν + k ν l + k l l=1 1 + l=1 q l+k 1 q ν l l + k ν k ν + k q l+k 1 q ν l. 2.4 l l + k ν + k q l+k 1 q ν l + 2 l + k ke ν+k/ Goig back to 2.4 we see that k 1 E 7 B + i 1 q k i=1 6 k i=1 1 ν + i. ν l=1 ν + k q l+k 1 q ν l l + k

7 It follows that 1 E X 1 X t 7 E X 1 X t 1 X t 1 + 1q 7 2 E X 1 X t 2 X t 2 + 1X t 2 + 2q t NN + 1 N + tq t 2.2 The upper boud We begi by provig some simple structural properties of G,p. Lemma 2.1. If θ > 1 ad θ σed σ 2θ 2e the w.h.p. there does ot exist S [], S σ such that es θ S. 2.5 Proof P S : S σ ad es θ S σ s=2θ σ s=2θ σ s θs 2 d 2.6 s θs θ s e eds s 2θ s θ 1 ed = e 2θ s=2θ d θ = O = o1 θ 1 θ s 2.7 provided d = o 1 1/θ. We will apply this lemma with θ 2 ε for ε 1 ad this fits with our boud o d. Lemma 2.2. Let σ, θ be as i Lemma 2.1. If 2θτ > 1 ad σed 2θτ 2θτ σ 4e the w.h.p. there do ot exist S T such that S = s σ, T τs ad d S v for every v T. 7

8 Proof I the light of Lemma 2.1, the assumptios imply that w.h.p. et : S \ T 2θτs. I which case, P S T, S σ, T τs : et : S \ T 2θτs σ s 2θτs s edt 2.8 s t 2θτ s=2θ t=τs σ s e s 2θτs eds 2s s 2θτ s=2θ t=τs σ s 2θτ s 2e = s eds 2θτ s=2θ t=τs σ s s 2θτ 1 2θτ s ed = 2e 2.9 2θτ s=2θ t=τs d 2θτ = O = o1. 2θτ 1 We will apply this lemma with 2θτ 2 ad this fits with our boud o d. Fix α > 12 ad let k = αd αd1 1/α ad β = ad γ = 16 l2 d l d l d αd. 1 1/α We will ow argue that w.h.p. A ca wi the game if k colors are available. A s iitial strategy will be the same as that described i [6]. Let C = C 1, C 2,..., C k be a collectio of pairwise disjoit subsets of [], i.e. a partial colorig. Let C deote k i=1 C i. For a vertex v let ad set Av, C = {i [k] : v is ot adjacet to ay vertex of C i }, av, C = Av, C. Note that Av, C is the set of colors that are available at vertex v whe the partial colorig is give by the sets i C ad v C. A s iitial strategy ca ow be easily defied. Give the curret color classes C, A chooses a ucolored vertex v with the smallest value of av, C ad colors it by ay available color. As the game evolves, we let u deote the umber of ucolored vertices i the graph. So, we thik of u as ruig backward from to 0. We show ext that w.h.p. every k-colorig proper or improper of the full vertex set has the property that there are at most γ vertices with less tha β/2 available colors. Let BC = {v : av, C < 3β/4}. 8

9 Lemma 2.3. W.h.p., for all collectios C, BC γ. Proof We first ote that if S = γ the w.h.p. S cotais at most 4γ 2 d edges. This follows from Lemma 2.1 with σ = γ ad θ = 4γd. It follows that for ay ε > 0 that there is a set S 1 S of size at least 1 εγ such that if v S 1 the its degree d S v i S is at most 8ε 1 γd. Fix C ad suppose that v S 1. Let bv, C = {i [k] : v is ot adjacet to ay vertex of C i \ S}. Thus av, C bv, C 8ε 1 γd. bv, C is the sum of idepedet idicator variables X i, where X i = 1 if v has o eighbors i C i \ S i G,p. The PX i = 1 1 p C i ad sice 1 p t is a covex fuctio of t we have Ebv, C k 1 p C i i=1 k1 p C C k /k k1 p /k = β oβ. It follows from the Cheroff boud 1.1 that Pbv, C 0.51β e β/33. Now, whe C is fixed, the evets {bv, C 0.51β}, v S 1 are idepedet. Thus, because av, C β/2 implies that bv, C 0.51β we have P C : BC γ k 1 εγ d e 1 εγ exp { = exp = o1, e 1 εγβ/33 { αd1 1/α l d + 1 εγ 33 l d l for large d ad small eough ε. } 1 εγ ε + l } α + 1 1/α l d 2 l l d 16 αd1 1/α 33 l d Let u 0 to be the last time for which A colors a vertex with at least β/2 available colors, i.e., u 0 = mi {u : av, C u 3β/4, for all v } C u, 9

10 where C u deotes the collectio of color classes whe u vertices remai ucolored. If u 0 does ot exist the A will wi. It follows from Lemma 2.3 that w.h.p. u 0 2γ ad that at time u 0, every vertex still has at least β/2 available colors. Ideed, cosider the fial colorig C i the game that would be achieved if A follows her curret strategy, eve if she has to improperly color a edge. Let U = {v / C u0 : av, C < β/2}. Now we ca assume that U γ. Because the umber of colors available to a vertex decreases as vertices get colored, from u 0 owards, every vertex colored by A is i U. Therefore u 0 2γ. Now let u 1 be the first time that there are at most 2γ ucolored vertices ad av, C u β/2, for all v C u. By the above, w.h.p. u 1 u 0, so i particular w.h.p. u 1 exists. A ca determie u 1 but ot u 0, as u 0 depeds o the future. A will follow a more sophisticated strategy from u 1 owards. We will show ext that we ca fid a sequece U = U 0 U 1 U l with the followig properties: The edges of U i : U i 1 \ U i betwee U i ad U i 1 \ U i will be divided ito two classes, heavy ad light. Vertex w is a heavy resp. light eighbor of vertex v if the edge v, w is heavy resp. light. P1 Each vertex of U i \ U i+1 has at most oe light eighbor i U i+1, for 0 i < l. P2 All U i : U i 1 \ U i edges are light for i 2. P3 Each vertex of U 1 either has degree at most β/3 i U 0 or it has at most β/20 heavy eighbors i U 0 \ U 1. P4 d Ui v β/3 for v U i \ U i+1. P5 U l cotais at most oe cycle. From this, we ca deduce that the edges of U 0 ca be divided up ito the heavy edges E H, light edges E L, the edges iside U l ad the rest of the edges. Assume first that U l does ot cotai a cycle. F = U, E L is a forest ad the strategy i [10] ca be applied. Whe attemptig to color a vertex v of F, there are ever more tha three F -eighbors of v that have bee colored. Sice there are at most β/3+β/20 o-f eighbors, A will succeed sice she has a iitial list of size β/2. If U l cotais a cycle C the A ca begi by colorig a vertex of C. This puts A oe move behid i the tree colorig strategy, i which case we ca boud the umber of F -eighbors by four. It oly remais to prove that the costructio P1 P5 exists w.h.p. Remember that d is sufficietly large here. We ca assume without loss of geerality that U 0 = 2γ. This will ot decrease the sizes of the sets av, U 0. 10

11 2.2.1 The verificatio of P1 P4: Costructig U 1 Applyig Lemma 2.2 with we see that w.h.p. σ = 2γ ad θ = d 1/α l 3 d ad = 2θ + β/4 < β/3 ad τ = θ/β U 1,a = {v U 0 : d U0 v 2θ + β/4} satisfies U 1,a 2τγ = 64d3/α l 6 d α 2 d 2. We the let U 1,a U 1,a be the subset of U 0 cosistig of the vertices with the 2τγ largest values of d U0. We the costruct U 1,b U 1,a by repeatedly addig vertices x 1, x 2,..., x r of U \U 1,a such that x j is the lowest umbered vertex ot i X j = U 1,a {x 1, x 2,..., x j 1 } havig at least three eighbors i X j. This eds with r 5 U 1,a i order that we do ot violate the coclusio of Lemma 2.1 with σ = 12τγ = 192d3/α l 6 d α 2 d 2 ad θ = 5/2 which is applicable sice It follows that 384ed 3/α l 6 d 5α 2 d 5/2 < 192d3/α l 6 d 2eα 2 d 2. U 1,b 12τγ Next let A be the set of vertices i U \ U 1,b that have two eighbors i U 1,b ad let B be the set of vertices i U 1,b that have more tha β/20 eighbors i A. We argue that w.h.p. B 1200d3/α τγ β = l4 d α 3 d3 6/α We first cosider the size of A. We first prove that w.h.p. A 12d 3/α τγ For a set S, let D 2 S deote the set of vertices ot i S that have at least two eighbors i S. Lemma 2.4. W.h.p. D 2 S d 3/α S for all S 12τγ. Proof For easy referece we ote that τγ = 40 l3 d α 2 d 2 2/α. For S 12τγ ad K = d 3/α, 11

12 we have P S 12τγ : D 2 S K S = 12τγ s=2 12τγ s=2 12τγ s Ks s d Ks e s e Ks s 2 d 2 s Ks 2 2 sed 2 K 1 e2 d 2 s=2 = o1. 2K Equatio 2.13 follows immediately from 2.11 ad Lemma 2.4. To boud the size of B we prove the followig lemma. Lemma 2.5. W.h.p. there do ot exist disjoit sets S, T such that T t 0 = 12d 3/α τγ ad S 100 T /β such that each v S has at least β/20 eighbors i T. 2K s Ks Proof We have P S, T deyig lemma = t 0 t=β/20 t 0 t=β/20 t 0 t=β/20 = o1. β/20 100t/β t d t 100t/β β/20 e t t t eβ 100t 100t/β 5t 20etd β 4 100/β e 100/β+6 β 100/β 5 d /β t Equatio 2.12 follows immediately from 2.13 ad Lemma 2.5. Now let A 1 be the set of vertices i A that have two eighbors of B. It follows from Lemma 2.4 that w.h.p. A 1 d 3/α B l4 d. α 3 d3 9/α Next let B 1 be the set of vertices i B that have at least β/20 eighbors of A 1. It follows from Lemma 2.5 that w.h.p. B A 1 β l5 d. α 4 d4 10/α Lemma 2.6. W.h.p. if s 0 = e d S s 1 = l5 d the NS d 1+1/α S. α 4 d 4 10/α 12

13 Proof P S : deyig lemma = s 1 s=s 0 s 1 s=s 0 = o1. s e s d 1+1/α s sd e d 1/α d1+1/α s d 1+1/α s Now let A 2 be the set of vertices i A 1 that have a eighbor i B 1. It follows from Lemma 2.6 that w.h.p. A l5 d. α 4 d3 11/α Note that if S < e d the Lemma 2.6 implies that NS S + d 1+1/α e d. Next let U 1,c = U 1,b ad Y 0 = A 2 B 1. We ow costruct U 1 U 1,c by repeatedly addig vertices y 1, y 2,..., y s of U \ U 1,c such that y j is the lowest umbered vertex ot i Y j = Y 0 {y 1, y 2,..., y j 1 } that has at least two eighbors i Y j. Takig θ = 3/2 ad usig Lemma 2.1, this eds with s 3 B by the same argumet used to show s 5 U 1,a above. Note that U 1 γ 1 = 13τγ = 208 l6 d α 2 d. 2 3/α We let W = U 0 \ U 1 ad partitio the edges W : U 1 ito light ad heavy edges. A W : Y s \ U 1,b : These edges will be heavy. B W \ A Y s : U 1,b. These edges will be light. Note that each v W \ A Y s has at most oe eighbor i U 1,b. C A \ A 1 Y s : U 1,b \ Y s. If v A \ A 1 Y s the v has at most two eighbors i U 1,b. At most oe of these ca be i B ad we make the correspodig edge light. If v has a eighbor i U 1,b \ B the we make the correspodig edge heavy. D A 1 \ Y s : U 1,b \ Y s. If v A 1 \ Y s the v has at most two eighbors i B. Noe of these ca be i B 1 ad we make the correspodig edges heavy. We ow have to check that P1 P4 hold. First cosider the light edges. There is at most oe for each v W ad so P1 holds. Now cosider the heavy edges. Vertices i B ca oly have light eighbors i W \ A ad vertices i B \ B 1 have at most β/20 heavy eighbors i A. Vertices i U 1,b \ B ca have at most β/20 heavy eighbors i W. Vertices i U 1 \ U 1,b have heavy degree at most β/3 i U 0. This verifies P3 ad P4 holds by the defiitio of U 1,a. 13

14 2.2.2 The verificatio of P1 P4: Costructig U 2 Applyig Lemma 2.2 agai, with σ = γ 1 ad θ = 3 ad = 2θ + β/3 ad τ = 12/β we see that w.h.p. U 2 = {v U 1 : d U1 v 6 + β/3} satsifies U 2 γ 2 = 12γ 1 β 2500 l7 d α 3 d3 4/α We the costruct U 2 U 2 by repeatedly addig vertices x 1, x 2,..., x r of U 1 \ U 2 such that x i has at least two eighbors i U 2 {x 1, x 2,..., x i 1 }. This eds with r 7 U 2 i order that we do ot violate the coclusio of Lemma 2.1 with which is applicable sice σ = 8γ l7 d 15 ad θ = α 2 d3 4/α e l 4 d 15α 3 d 2 4/α 15/8 < l7 d 2eα 3 d 3 4/α This verifies P1 P4 with i = The verificatio of P1 P5: Costructig U i, i 3 We ow repeat the argumet to create the sequece U 0 U 1 U l. The value of θ has decreased to 15/8 see 2.15 ad U i 12γ/β U i 1, as i We choose l so that U l l. We ca easily prove that w.h.p. S cotais at most S edges wheever S l, implyig P5. This completes the proof of Part b of Theorem Theorem 1.3: G,d We will ot chage A or B s strategies. We will simply trasfer the relevat structural results from G,d/ to G,d. Some of the uimportat costats will chage, but this will ot chage the verificatio of the success of the various strategies. We will first do this usig Theorem 1.2 uder the assumptio that d 1/4. For larger d we will use Theorem 1.1 ad the sadwichig theorem of Kim ad Vu [15]. This latter aalysis is give i Sectio

15 3.1 d 0 d 1/4 Here we assume that d 0 is a sufficietly large costat. We begi with the cofiguratio model of Bollobás [8]. We have a set W of poits ad this is partitioed ito sets W 1, W 2,..., W of size d. We defie φ : W [] by φx = j for all x W j. We associate each pairig or cofiguratio F of W ito W /2 pairs to a multigraph G F o the vertex set []. A pair d! d/2!2 d/2 {x, y} F becomes a edge φx, φy of G F. Now there are pairigs ad each simple d-regular graph without loops or multiple edges arises d! times as G F. So for ay pair of d-regular graphs G 1, G 2 we have PG F = G 1 G F is simple = PG F = G 2 G F is simple. 3.1 I order to use this, we eed a boud o the probability that G F is simple. This is the cotet of Lemma 2 of [9]. It follows from 3.1 ad 3.2 that for ay graph property A: PG F is simple e 2d ε 2d2 PG F A = o1 implies PG,d A = o We ca use the above to estimate ρ = PG,d/ is d regular. We write this as ρ = PG = G,d/ is d regular EG = d/2p EG = m = d/2. It is easy to show, usig Stirlig s approximatio, that P EG = m = Ωm 1/2 ad so we cocetrate o the other factor. Let N = 2. There are N m Ne m m, graphs with vertex set [] ad m edges of which e 2d2 d! Ω are d-regular. d/2!2 d/2 d! So, sice d = o, ρ = Ω e 2d2 d 1/2 d/2 d e 1 d! d/2 d = e 1 Ω d d d 1/2 e d+2d2 d! = Ω / d We eed aother crude estimate. We prove a small modificatio of Lemma 1 from [9]. 15

16 Lemma 3.1. Give {a i, b i }, i = 1, 2,... k /8d the k 20d Pa i, b i EG,d, 1 i k. Proof let Let G d deote the set of d-regular graphs with vertex set []. For 0 t k we Ω t = {G G d : {a i, b i } EG, 1 i t ad {a i, b i } / EG, t + 1 i k}. We cosider the set X of pairs G 1, G 2 Ω t Ω t 1 such that G 2 is obtaied from G 1 by deletig disjoit edges {a t, b t }, {x 1, y 1 }, {x 2, y 2 } ad replacig them by {a t, x 1 }, {y 1, y 2 }, {b t, x 2 }. Give G 1, we ca choose {x 1, y 1 }, {x 2, y 2 } to be ay ordered pair of disjoit edges which are ot icidet with {a 1, b 1 },..., {a k, b k } or their eighbours ad such that {y 1, y 2 } is ot a edge of G 1. Thus each G 1 Ω 1 is i at least D 2kd 2 + 1D 2kd pairs, where D = d/2. Each G 2 Ω t 1 is i at most 2Dd 2 pairs. The factor of 2 arises because a suitable edge {y 1, y 2 } of G 2 has a orietatio relative to the switchig back to G 1. It follows that Ω t Ω t 1 2Dd 2 D 2kd 2 + 1D 2kd 2 + 2d d. It follows that ad this implies the lemma. Ω k Ω Ω k 20d k The lower boud Usig 3.2 we ca replace 2.3 by e 2d2 exp { β + γ + β 2 2β + γ 2 /2 + o d 1d 1 l 2 d } = o1 for d 1/4. After this, we ca argue as i the case G,p The upper boud We first eed to prove the equivalet of Lemmas 2.1 ad 2.2. Lemma 3.2. If 1 < θ d 1/6 l 3 d ad 10σed θ θ σ 2e 3.5 the w.h.p. there does ot exist S [], S σ such that es θ S. 16

17 Proof where P S : S σ ad es θ S π s = max X [s] 2 X =θs σ s=2θ PEG,d X. s s 2 θs π s It follows from 3.2 that π s e 2d2 d θs. If d is small, say d l 1/3 the we ca see from the proof of Lemma 2.1 that P S : S σ ad es θ S O e 2 l2/3 d θ = o1. θ 1 We ca therefore assume that d l 1/3 ad the σ s σ 2 s π s e 2 2d2 s θs s θs s=3d 2 s=3d 2 σ s e 2d2 e e 2d2 = o1. s=3d 2 σ s=3d 2 2 s d θ 1 ed 2θ θs θ s Whe s 3d 2 we use Lemma 3.1. For this we will eed to have θs 3θd 2 /8d. The maximum value of θ is d 1/6 l 3 d ad so the lemma ca ideed be applied for d 1/4. Assumig this, we have 3d 2 2θ s s 2 θs π s 3d 2 s 2 s θs 2θ 3d 2 s e 2θ = o1. Lemma 3.3. Let σ, θ be as i Lemma 3.2. If 2θτ 10σed σ 2θτ 4e 20d θs θ 1 θ s 20ed 2θ the w.h.p. there do ot exist S T such that S σ, T τs ad d S v for v T. 17

18 Proof We first argue that if d l 1/3 the we prove the lemma by just iflatig the failure probability by e 2d2 as we did for Lemma 3.2. We therefore assume that d l 1/3 ad write P S T, S σ, T τs, et : S \ T 2θτs s,t s s t ts t 2θτs π s where ow we have Usig 3.2 we write σ s s=3d 2 /τ t=τs σ e 2d2 e 2d2 e 2d2 = o1. s π s = s t s=3d 2 /τ t=τs σ s=3d 2 /τ t=τs σ s 2 s s=3d 2 /τ t=τs max PEG,d X. X T S\T X = 2θτs ts t 2θτs π s s s edt s t 2θτ s s 2θτ 1 2e 2θτs ed 2θτ 2θτ s Whe s 3d 2 /τ use Lemma 3.1, with the same caveats o the value of d. So, 3d 2 /τ s s=2θ t=τs s s t s ts t 2θτs 3d 2 /τ s s ts t 20d t 2θτs s=2θ t=τs d 2θτ = O = o1. 2θτ 1 π s 2θτs Remark 1. We ca estimate P C : BC γ by multiplyig 2.10 by 1/ρ ad otice that it remais o1. After this, the proof will much the same as for G,p, but with a few costats beig chaged. 18

19 3.2 1/4 d 1/3 ε Our approach i this sectio is to use the sadwichig techique developed by Kim ad Vu i [15] to adapt the proof of Theorem 1.1. I some sese it is pretty clear that give the results of [15], it will be possible to traslate the results of [6] to deal with large regular graphs. We will carry out the task, but our proof will be abbreviated ad rely o otatio from the latter paper. Without chagig the strategy used i obtaiig the lower boud, we show that each itermediate result used to prove the theorem i [6] cotiues to hold for radom regular graphs G,d i the rage where these ca be approximated sufficietly well by radom graphs G,d/. I order to get the required stregth from the Kim-Vu couplig, however, we require d = p = ε for some ε ε 0, where ε 0 is a small absolute costat Notatio We use Theorem 2 i [15] to get a joit distributio o H 1, G, H 2 : G is d-regular, H 1 G, H 1 H 2, ad although G H 2, this is almost true i a way we discuss further. The graphs H 1 ad H 2 are radom graphs with edge probabilities p 1 ad p 2, ad by judicious choice of parameters we ca set p 1 = p/1 + δ ad p 2 = p1 + δ, where l p = d 1/3 ad δ = Θ. d Costats defied i [6] are i terms of p ad we will make this relatioship explicit. Of ote is the costat l 1 p = log b log b log b p 10 log b l where b = bp = 1 1 p Kim-Vu couplig The costructio of the couplig H 1, G, H 2 i [15] yields H 1 G w.h.p., but ot G H 2. As a substitute for such a result, we prove the followig lemma. Lemma 3.4. G \ H 2 = O1 w.h.p. Proof We rely o the boud G\H 2 G δh 2 + H 2 \G. Trivially, G = d. Part 3 of Theorem 2 i [15] states that w.h.p. H 2 \ G 1 + o1 l lδd/ l 1 + o1 l 3 + o1 = 2 l d 2 =. l l + O1 2ε

20 We prove that w.h.p. δh 2 d. For ay vertex v, deg H2 v follows the biomial distributio B 1, p 2. By the Cheroff boud, [ ] δ P[deg H2 v < d] P B 1, p 2 < 1 1p 2 e δ2 /101+δ δ We ca simplify the expoet here to δ2 d δ = Ωl 2/3 d 1/3 Ω ε/ δ So P[deg H2 v < d] O Ωε/3 ad P[δH 2 < d] = o1, completig the proof Bouds We first prove a few auxiliary bouds o the relatioship betwee p, p 1, ad p 2, as well as other costats i terms of these probabilities. Boud l 1p l 1 p 1 1 2δ, ad 1 l 1p 1 l 1 p 1 + 2δ. Proof let x = We first ote that if p the the derivative log bp p < 0 ad so if we the, log p p l 10 l 1 p 1 l 1 p = log bp 1 log bp1 log bp1 p 1 10 log bp1 l log bp log bp log bp p 10 log bp l log bp 1 log bp1 log bp p 10 log bp1 l log bp log bp log bp p 10 log bp l = log bp 1 x log bp x = l bp l bp 1 p1 + p p δ. This proves the secod iequality. For the first, we take the reciprocal, ad ote that 1 + 2δ 1 > 1 2δ. Boud l 1p 2 l 1 p 1 2δ, ad 1 l 1p l 1 p δ. Proof Apply Boud 3.1 with p 2 i place of p ad p i place of p 1, sice their relatioships are the same. Note ow that if d = θ where θ = Θ1 the which implies that log b log b p log b = l log b p l 1 θ l 1 p = θ + o1 log b p

21 Boud p 1 l 1p = l 1pl 10 θ + o1 ad 1 p 2 l 1p = l 1pl 10 θ + o1. Proof. It follows from Boud 3.1 that 1 p 1 l1p = 1 + o11 p l1p = 1 + o1 log b plog 10. Now use 3.6. The proof for p 2 is similar Lemmas used for the lower boud i [6] The strategy used i [6] to prove the lower boud relies o probabilistic assumptios labeled there as Lemmas 2.1 through 2.4. By assumig that those lemmas hold for radom graphs ad occasioally referecig the proofs of the origial lemmas, we prove that they hold i the radom regular case as well. It follows that the lower boud of Theorem 1.1 is valid i the case of G,d as well, provided our assumptio that d = ε holds. Lemma 3.5 Lemma 2.1 of [6]. For every S [] with S = l 1 p, w.h.p. l 1 pl 9 NS l1 pl 11. Proof. For a S as above, NS = N G S N H1 S. The distributio of N H1 S is biomial with mea 1 p 1 l 1p, which is at most Ol 1 pl 10 by Boud 3.3. We ca use Cheroff bouds to get N H1 S l 1 pl 11, which implies the same for NS. The proof of the lower boud is similar, except that we do t have the strict cotaimet N H2 S N G S. However, by Lemma 3.4, ay vertex i S has O1 eighbors i G that it does ot have i H 2. Therefore N G S N H2 S + O S. Because S = l 1 p, ad the Cheroff boud gives N H2 = Ωl 1 pl 10 w.h.p., this differece will be absorbed i the 1 + o1 asymptotic factor. Lemma 3.6 Lemma 2.2 of [6]. W.h.p. there do ot exist S, A, B such that coditios omitted ad every x B has fewer tha ap/2 eighbors i A where a = A. Proof. The proof of the correspodig lemma i [6] relies o the distributio to say that the umber of eighbors ay x B has i A is distributed accordig to the biomial distributio Ba, p, ad uses the Cheroff boud P[Ba, p ap/2] b 1 e ab 1p/8. The umber of edges betwee x ad A is bouded below by the umber of such edges i the graph H 1, which is distributed accordig to Ba, p 1. So we replace the boud above by [ P[Ba, p 1 ap/2] = P Ba, p 1 ap ] 11 + δ P [Ba, p 1 ap δ ]. 2 2 By the Cheroff boud, this is at most e ap 11 δ 2 /8 e 1 o1ab 1p/8, ad the argumet of [6] still goes through. 21

22 Lemma 3.7 Lemma 2.3 of [6]. Let a 1 = 2000ε 2 where ε is a small positive costat. W.h.p. there do ot exist sets of vertices S, T 1,..., T a1 such that coditios omitted ad NS T i = for i = 1,..., a. Proof. If such sets exist i the graph G, the they will still exist whe we lose some edges i passig to the graph H 1. Examiig the proof i [6] we see that all it requires is to cosider the followig factor which iflates the o1 probability estimate they use by: 1 p1 a1 εl 1 p/ p 3l1 p 1 2 p1 = p 3l1 p pδ 2 = 1 + o δ1 p Lemma 3.8 Lemma 2.4 of [6]. Let t = l 1. W.h.p. there do ot exist pairwise disjoit pl 7 sets of vertices S 1,..., S t, U, such that coditios omitted ad U NS i l 1 pl 8 for i = 1,..., t. Proof. Suppose such sets exist i the graph G. By Lemma 3.4 each vertex of S i has at most O1 eighbors i G that are ot i H 2 ; therefore i passig to the graph H 2, U NS i will be at most l 1 pl 8 + Ol 0 p where each S i = l 0 p = l 1 p + C/p for some costat C a fact we will use agai. Sice l 1 p = l, the size of U NS θ+o1p i i H 2 is also 1 + o1l 1 pl 8. There is room i the argumet of [6] to prove this lemma for itersectios of this size as well. Therefore we will proceed by arguig that w.h.p. sets such as S 1,..., S t, U do ot exist i H 2. The proof i [6] higes upo the claim that 1 p l 0p = Ω1 p l 1p. So it suffices to prove that 1 p 2 l 0p = Ω1 p l 1p. We split 1 p 2 l 0p ito factors 1 p 2 l 1p ad 1 p 2 C/p. By Boud 3.3, the first factor is Ω1 p l 1p. The secod factor is o less tha 1 p 2 C/p 2, which stays i e C, 4 C ] as p 2 rages over 0, 1/2], so it is effectively a costat Lemmas used for the upper boud i [6] As i the lower boud, the strategy used i [6] to prove the upper boud relies o some properties that hold w.h.p. i G,p. Agai, we apply the Kim-Vu Sadwich Theorem [15, Theorem 2] to show that the same properties hold i G,d as well. The first player s strategy described i [6] is simple she chooses a ucolored vertex with miimal umber of available colors ad the colors it with a arbitrary available color. We preset some otatio used i [6] durig the aalysis of the strategy. Give a partial colorig C ad a vertex v let αv, C be the umber of available colors for v i C. For a costat α > 3 defie β G = α p 1/α log b p, γ G = l β G

23 ad BC = {v αv β G /2}. The first lemma states that there are ot may vertices with few available colors. We show that the same is also true i G,d. Lemma 3.9 Lemma 3.1 of [6]. W.h.p. for all collectios C, BC γ. Proof Here we ca just use Remark 1. Lemma 3.10 Lemma 3.2 of [6]. W.h.p. every subset S of G,p of size s spas at most φ = φs = 5ps + l s edges. Proof. The proof i [6] actually gives the result for φ 2 = 4.5ps l s. Thus, applyig Lemma 3.2 of [6] to H 2 gives that w.h.p. every set S of size s spas at most 4.5p 2 s+0.9 l s edges. Sice w.h.p. every vertex of G touches at most O1 edges ot i H 2, we have that w.h.p. the umber of edges spaed by S i G is bouded by 4.5p 2 s l + O 1 s = 4.5ps1 + o l + O1s 5ps + l s as required. This completes the proof of Theorem Theorem 1.4: Radom Cubic Graphs Cosider the colorig game o G,3 with three colors. We describe a strategy for B that wis the game for him w.h.p., so χ g G,3 4 w.h.p. This proves Theorem 1.4: i geeral χ g G G + 1 where deotes maximum degree, ad i particular χ g G,3 4. We proceed i two steps. First, we describe a strategy for B that wis the game, give the existece of a subgraph H i G satisfyig certai coditios. Next, we will prove that w.h.p., a radom cubic graph cotais such a subgraph. 4.1 The wiig strategy We will say that two vertices are close if they are coected by a path of legth two or less, ad that a path is short if some vertex o it is close to both edpoits. This is ot the same as beig of legth at most four. Vertices that are ot close are far apart ad a path that is ot short is log. The motivatio for this termiology is that colorig a vertex ca oly have a effect o vertices that are close to it; we will make this precise later o. We first assume the existece of a subgraph H with the followig properties see Figure 1: 23

24 w Figure 1: The subgraph H required for Bob s strategy o Radom Cubic Graphs. 1. H cosists of two vertices, v ad w, together with three iterally disjoit paths from oe to the other. 2. Each of the paths cosists of a eve umber of edges. 3. No two vertices i H are coected by a short path outside of H i particular, H is iduced. 4. The three paths themselves are all log. I additio, Property F: if A goes first, the the vertex colored by A o her first move is far from H. B first plays o the vertex v. Provided A s ext move is ot o the vertex w, or o the eighbors of v or w, it is close to at most oe of the three paths which make up H this follows from Properties 3 ad 4. The remaiig two paths form a cycle cotaiig v, with o other already colored vertices close to the cycle; by Property 2, the cycle is eve. Call the vertices aroud the cycle v, v 1, v 2,..., v 2k 1. Startig from this eve cycle, B proceeds as follows. He colors v 2 a differet color from v; this creates the threat that o his ext move, he will color the third eighbor of v 1 the remaiig color, leavig o way to color v 1 ad wiig. We will call such a move by B a forcig move at v 1. A ca couter this threat i several ways: By colorig v 1 the oly remaiig viable color. By colorig v 1 s third eighbor the same color as either v or v 2. 24

25 By colorig that eighbor s other eighbors i the color differet from both v ad v 2. I all cases, A must color some vertex close to v 1, that does ot lie o the cycle. B cotiues by makig a forcig move at v 3 : colorig v 4 a differet color from v 2. Cotiuig to play o the eve vertices v 2i, B makes forcig moves at each odd v 2i 1. By Property 3 of H, the set of vertices A must play o to couter each threat are disjoit; thus, A s respose to each forcig move does ot affect the rest of the strategy. By Property F, A s first play does ot affect the strategy either. Whe B colors v 2k 2, this is a forcig move both at v 2k 3 ad at v 2k 1 provided Bob chooses a color differet both from v 2k 4 ad v. A caot couter both threats, therefore B wis. We ow accout for the remaiig few cases. If A colors a eighbor of v or w o her secod move, this vertex will be close to all three possible eve cycles. However, we kow that all three paths i H have eve legth. Therefore we ca still apply this strategy to the eve cycle ot cotaiig the vertex A colored. Eve though it will be close to v or w, we will ever eed to force at v or at w, because we oly force at odd umbered vertices alog the path. Fially, if A colors w itself, the there is o path we ca choose that will avoid the vertex. Istead, B picks ay of the paths from v to w, ad makes forcig moves dow that path. Provided that the path is sufficietly log to do so which follows from Property 4, the fial move will be a forcig move i two ways, wiig the game for B oce agai. 4.2 Proof of the existece of H It remais to show that the subgraph H exists w.h.p. eve allowig for A s first move. We will assume G is chose by addig a radom perfect matchig to a cycle C o vertices, ad fid H w.h.p. That this is a cotiguous model to G,3 is well kow, see [17]. I the followig, let c be a costat; we will later see that we eed c to be less tha 1 for the proof to hold. We begi by coutig good segmets of legth m = c o C, by which we mea those with o iteral chords. First of all let X be twice the umber of chords that itercept segmets of legth m or less these are the oly chords that could possibly be iteral to a segmet of the desired legth. X ca be writte as the sum X 1 + X 2 + X, where X i is the 0-1 idicator for the i-th vertex call it v i to be the edpoit of such a chord. Also, let Y i deote the legth of the smaller of the two segmets defied by v i this segmet stretches from v i to its parter. Thus PY i = t = { t 1/2 t = /2, eve 25

26 Clearly X i = 1 if ad oly if Y i m, ad so EX i = 2m 1 I additio, VarX i EX i, ad so ad EX = 2m 1 2c. VarX = CovX i, X j EX + i j CovX i, X j. i=1 VarX i + i j Now CovX i, X j = 4m2 1 + m PX 2 i = 1 Y j = tpy j = t t=2 4m m 2 3 2m 1 8m 2 = Thus, By Chebyshev s iequality, VarX EX + 8m EX. P X EX λex VarX λ 2 EX 2 2 λ 2 c. Puttig λ = 1/5 we see that w.h.p. X 2c. Cosider the differet segmets of legth m o C. Each chord couted by X elimiates at most m of these segmets as beig good, which leaves 1 c 2 segmets remaiig. We will wat o-overlappig good segmets; each good segmet overlaps at most 2m other good segmets ad so we ca assume that we ca fid 2 1 c 1 c /2 o-overlappig good segmets w.h.p. Here the segmets are σ 1, σ 2,..., σ 21 are i clockwise order aroud C. We pair them together P i = σ i, σ 1 +i, i = 1, 2,..., 1. Pick ay pair P j. If there are exactly 3 chords from oe segmet to the other, as i Figure 2, the we will costruct H as follows assumig a i ad b i are the edpoits of the chords, as labeled i Figure 2: Set v ad w to be a 2 ad b 2, respectively. The first path from v to w is a 2,..., a 1, b 3,..., b 2, where the vertices i the ellipses are chose alog C. The secod path from v to w is a 2, b 1,..., b 2. 26

27 a 1 a 3 b 1 2 b 3 Figure 2: A typical example of the subgraph H foud i the Hamiltoia cycle model. The third path from v to w is a 2,..., a 3, b 2. The paths give above require that the three chords are a 1, b 3, a 2, b 1, ad a 3, b 2. I order for H to satisfy Properties 2 ad 4, we impose coditios o the legths of the paths a 1,..., a 2, a 2,..., a 3, b 1,..., b 2, ad b 2,..., b 3 : they must ot be too small, ad must have the right parity so that the three paths from v to w have eve legth. However, these coditios ad the coditio that a 2, b 2 must ot be a chord elimiate oly a costat fractio of the possible chords; therefore there are Ωm 6 ways to choose the chords. The probability, the, that a subgraph H ca be foud betwee two give good segmets, is at least m 6 Ω 1 m 2m m where the last factor bouds the probability that there are o extra chords betwee the two segmets. This teds to a costat ζ 1 that does ot deped o. Thus the expected umber of j for which P j has Properties 1,2 ad 4 are satisfied is at least ζ 1 1. We ow cosider the umber of pairs of good segmets i which we ca hope to fid this structure. I order to esure that, should a subgraph H be foud, it satisfies Property F, we elimiate all pairs which cotai a vertex close to the vertex A chooses o her first move a costat umber of pairs. 27

28 To esure Property 3 we elimiate all pairs P j i which two vertices have chords whose other edpoits are 1 or 2 edges apart. This happes with probability O1/ for ay two vertices, regardless of the dispositio of the other chords icidet with the segmets i P j. The pair P j s cotais 2m 2 2c 2 pairs of vertices. Therefore with probability at least 1 O1/ 2c2, which teds to a costat, ζ 2 say, a pair P j satisfies Property 3. Thus the expected umber of j for which the pair P j give rise to a copy of H satisfyig all required properties is at least ζ 1 where ζ = ζ 1 ζ 2. To prove cocetratio for the umber of j we ca simply use the Chebyshev iequality. This will work, because exposig the chords icidet with a particular pair P j will oly have a small effect o the probability that ay other P j has the required properties. Refereces [1] D. Achlioptas ad C. Moore, The chromatic umber of radom regular graphs, I Proceedigs of the 7th Iteratioal Workshop o Approximatio Algorithms for Combiatorial Optimizatio Problems ad 8th Iteratioal Workshop o Radomizatio ad Computatio, volume 3122 of Lecture Notes i Computer Sciece, Spriger [2] D. Achlioptas ad A. Naor, The two possible values of the chromatic umber of a radom graph, Aals of Mathematics [3] N. Alo ad J. H. Specer, The probabilistic method, 3 rd Ed., Wiley, New York, [4] T. Barticki, J.A. Grytczuk, H.A. Kierstead ad X. Zhu, The map colorig game, to appear i America Mathematical Mothly. [5] H.L. Bodlaeder, O the complexity of some colorig games, Iterat. J. Foud. Comput. Sci , [6] T. Bohma, A.M. Frieze ad B. Sudakov, The game chromatic umber of radom graphs, Radom Structures ad Algorithms [7] B. Bollobás, The chromatic umber of radom graphs, Combiatorica , [8] B. Bollobás, A probabilistic proof of a asymptotic formula for the umber of labelled regular graphs, Europea Joural o Combiatorics [9] C. Cooper, A.M. Frieze ad B. Reed, Radom regular graphs of o-costat degree: coectivity ad Hamilto cycles, Combiatorics, Probability ad Computig

29 [10] U. Faigle, U. Ker, H. Kierstead ad W.T. Trotter, O the Game Chromatic Number of some Classes of Graphs, Ars Combiatoria [11] A.M. Frieze ad T. Luczak, O the idepedece ad chromatic umbers of radom regular graphs, Joural of Combiatorial Theory B [12] M. Garder, Mathematical Games, Scietific America , o. 4, [13] S. Jaso, T. Luczak ad A. Ruciński, Radom Graphs, Joh Wiley ad Sos, New York, [14] G. Kemkes, X. Pérez-Giméez ad N. Wormald, O the chromatic umber of radom d-regular graphs, Advaces i Mathematics , [15] J.H. Kim ad V.H. Vu. Sadwichig radom graphs: uiversality betwee radom graph models. Advaces i Mathematics : [16] T. Luczak, The chromatic umber of radom graphs, Combiatorica , [17] N. Wormald, Models of radom regular graphs, Surveys i Combiatorics, J.D. Lamb ad D.A. Preece, eds., Lodo Mathematical Society Lecture Note Series, Cambridge Uiversity Press, Cambridge,