The t-tone chromatic number of random graphs

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1 The t-toe chromatic umber of radom graphs Deepak Bal Patrick Beett Adrzej Dudek Ala Frieze March 6, 013 Abstract A proper -toe k-colorig of a graph is a labelig of the vertices with elemets from ( [k] such that adjacet vertices receive disjoit labels ad vertices distace apart receive distict labels. The -toe chromatic umber of a graph G, deoted τ (G is the smallest k such that G admits a proper -toe k colorig. I this paper, we prove that w.h.p. for p C 1/4 l 9/4, τ (G,p = ( + o(1χ(g,p where χ represets the ordiary chromatic umber. For sparse radom graphs with p = c/, c costat, we prove that τ (G,p = ( / where represets the maximum degree. For the more geeral cocept of t-toe colorig, we achieve similar results. 1 Itroductio The ordiary chromatic umber of a graph G, deoted χ(g is the fewest umber of colors ecessary to label the vertices of G such that o two adjacet vertices receive the same color. There have bee may geeralizatios of this cocept, for example list colorig, t-set colorig [4], ad distace-t colorigs [5]. A atural extesio which further geeralizes the cocepts metioed above is that of a t-toe colorig. Chartrad itroduced t-toe colorig as a geeralizatio of proper colorig, which is equivalet to 1-toe colorig. The D. Bal, P. Beett ad A. Frieze Departmet of Mathematical Scieces Caregie Mello Uiversity Pittsburgh, PA {dbal, ptbeet, af1p}@adrew.cmu.edu A. Dudek Departmet of Mathematics Wester Michiga Uiversity Kalamazoo, MI adrzej.dudek@wmich.edu

2 Deepak Bal et al. cocept was iitially studied i a research group directed by Zhag [8] ad the ivestigated by Bickle ad Phillips [1]. Throughout the paper, if l k are positive itegers, [k] refers to the set {1,,..., k} ad ( [k] l refers to the collectio of l sized subsets of [k]. For vertices u ad v of G, d(u, v refers to the distace betwee u ad v, i.e. the miimum umber of edges o a path betwee u ad v. We may ow give the formal defiitio of a t-toe colorig which appears i [8] ad [1]. Defiitio 1 Let G = (V, E be a graph ad let t be a positive iteger. A (proper t-toe k-colorig of a graph is a fuctio f : V (G ( [k] t such that f(u f(v < d(u, v for all distict vertices u ad v. A graph that admits a t-toe k-colorig is t-toe k-colorable. The t-toe chromatic umber of G, deoted τ t (G is the least iteger k such that G is t-toe k-colorable. For a vertex v V (G, we call f(v the label o v. The elemets of f(v are colors. I this paper, we are cocered primarily with the -toe chromatic umber, τ (G. Note that the defiitio i this case says that adjacet vertices receive disjoit labels ad vertices at distace receive distict labels. The classical Erdős-Réyi-Gilbert radom graph G,p is a graph o vertex set [] i which each potetial edge i ( [] appears idepedetly with probability p. We say that a evet occurs with high probability, deoted w.h.p., if the probability of the evet teds to 1 as teds to ifiity. The mai results of this paper cocer τ (G,p i two rages of p. O the dese ed of the spectrum, we have the followig result. Theorem 1 Let p = p( satisfy C 1/4 l 9/4 p < ε < 1 where C is a sufficietly large costat ad ε is ay costat < 1. The w.h.p., τ (G,p = ( + o(1χ(g,p. For sparse radom graphs we prove the followig: Theorem Let c be a costat, ad let p = c/. If we let represet maximum degree, the w.h.p., τ (G,p =. I the dese rage w.h.p., the diameter of G,p is. Thus fidig a t-toe colorig of the radom graph i this rage amouts to fidig labels which are disjoit o adjacet vertices ad itersect i at most oe color o o-adjacet pairs. For this reaso, our proof techiques for the t = case may be easily exteded to the t 3 case. Our result i the sparse case relies o aother tight result for -toe colorigs of trees. There is o kow aalogous tight result for t-toe colorigs with t 3. Hece the result we have i the sparse case for t 3 is weaker. The results for t 3 appear as Theorems 3 ad 4 i Sectio 5.

3 The t-toe chromatic umber of radom graphs 3 A lower boud o τ t (G Cosider a t-toe k-colorig of ay graph G o vertex set [], ad for each i [k], let S i be the set of vertices that have color i as oe of their t colors. Whe we sum S i over i, each vertex is couted t times (oce for each color it has. Thus so τ t (G t = i S i k α (G t α(g. The above iequality, together with kow bouds o χ (G,p /α (G,p give us a lower boud o the t-toe chromatic umber of G,p. I particular, w.h.p.. τ t (G,p (t o(1χ(g,p 3 Upper boud for dese case Throughout this sectio, let G = G,p o vertex set [] ad let C 1/4 l 9/4 p < 1 where C is a sufficietly large costat. We adapt the proof strategy of Bollobás [3] for obtaiig bouds o the ordiary chromatic umber χ(g. Bollobás strategy requires two key facts. First, oe shows that w.h.p. every sufficietly large subgraph has a idepedet set almost as large as α (G,p. The we show that w.h.p. there are o small subgraphs with high edge desity. The strategy for colorig G,p is as follows: iteratively fid a maximum idepedet set i the graph ad remove the vertices, util the remaiig set of vertices is sufficietly small. The remaiig graph does ot have high edge desity. Thus we may greedily color the rest of the vertices usig ew colors (ad ot very may of them. W.h.p. the resultig colorig uses (1 + o(1 = α (G,p log 1 1 p (1 + o(1 colors, which is clearly asymptotically optimal. If we wat a -toe colorig, we may begi by givig G,p a ordiary proper colorig as above. But the we have to assig each vertex aother color, ad the colors we assig i this secod pass must be carefully chose with regard to the colors that are already there. From the lower boud, we kow that we will eed to use at least (roughly twice the umber of colors we would eed for a ordiary colorig. So for our secod pass we might as well use ew colors (i.e. oe of the same colors we used i the first pass.

4 4 Deepak Bal et al. Also, w.h.p. the diameter of G,p is for this rage of p, so i our fial -toe colorig we caot assig ay two vertices the same pair of colors. Let P = {P 1, P,..., P a }, R = {R 1, R,..., R b } be partitios of []. We will say that a set R respects P if R i P j 1 for all i, j. We also say that a specific set R respects P if R P j 1 for all j. We ca fid a -toe colorig of G,p by fidig two ordiary colorigs such that the partitios geerated by the color classes respect each other. We will accomplish that task with the proof strategy discussed above i mid. Start with a partitio P of [] ito sets of vertices that are idepedet i G,p (i.e. the parts of P are color classes of a proper colorig. The iteratively fid large idepedet sets that respect P ad remove those vertices from the graph. Oce the remaiig graph is sufficietly small, it has low eough edge desity to be colored greedily usig all ew colors without havig a sigificat impact o the total umber of colors used. For ease of otatio, set b := 1 1 p, ad set k := 3 log b = 3 l l b. The followig bouds are well kow (see, e.g., [3] or [9]: w.h.p. α (G,p < k ad χ (G,p < 3 k. The two key lemmas we will require to prove Theorem 1 are as follows: Lemma 1 W.h.p. for every partitio P of [] ito at most 3 k parts of size at most k, ad every set U [] of size U >, G[U] cotais a idepedet set of size at least s 0 := log b log b log b 5 log b l = l which respects P. l + l l b 7 l l l b Lemma W.h.p. for every set H [] of size at most, G[H] has at l most H k l edges. Assumig the truth of the lemmas, the proof of Theorem 1 is as follows: Proof (Theorem 1 We start with a ordiary colorig of the vertices usig (1 + o(1 χ (G,p colors, where the partitio P give by the color classes has at most 3 k parts, ad each part is of size at most k. W.h.p. such a colorig exists. Now we apply Lemma 1 iteratively, fidig large idepedet sets that respect P ad removig the idepedet sets from the graph, util less tha l vertices remai. Let V be the set of vertices remaiig at this poit. We will use a ew set of colors for the vertices of V. All we have to do is make sure that the color classes withi V respect the partitio P. Thus, the problem of colorig V is equivalet to fidig a (ordiary colorig of the graph G with vertex set V, ad with the edge set beig the uio of E (G,p [V ] ad the set of edges with both edpoits i the same part of P. The latter set of edges guaratees that o two vertices i the same part of P will be assiged the same color. Thus, ay proper ordiary colorig of G will serve as a valid completio of our -toe colorig of G,p. Now the chromatic umber of G is at most its colorig umber (see, e.g., Prop. 5.. i [7], which is at most { E(G } [H] 1 + max : H V H

5 The t-toe chromatic umber of radom graphs 5 where E(G represets the edge set of a graph G. But by Lemma, w.h.p. for all H V we have E(G,p[H] H k l. Of course, G [H] has some edges that are ot i E(G,p [H]. Specifically, G [H] has all possible edges with both edpoits i the same part of P. Usig Jese s iequality, the covexity of the fuctio ( x, ad the properties of P, we see that G [H] has at most H ( k k = O( H k such edges. Therefore for ay H V, we have E(G [H] H k l + O(k = o(χ(g,p. Thus we ca color G usig a egligible umber of colors. We ow prove the two lemmas. Proof (Lemma 1 Fix P = {P 1, P,..., P m }, a partitio of [] ito m < 3 k parts of size at most k, ad let the radom variable X be the um- of idepedet sets of size s := log b log b log b log b l = ber l + l l b 3 l l l b respectig P. Note that s > s0 from the statemet of Lemma 1. By a applicatio of Jaso s iequality (see, e.g., iequality.18(ii i [9], ( E[X] P [X = 0] exp S,S E [X (3.1 SX S ] where the sum i the deomiator is take over all pairs of sets of vertices S, S of size s such that S S ad S, S respect P. The radom variable X S is just a 0 1 idicator for whether S is idepedet i G,p. Note that the umber of s-sets ot respectig P is at most so 1 i m ( ( Pi s EX ( = O ( ( s ms 3 = O s ( ( (1 p (s 1 O s ( s 3. ( s Now we would like to put a upper boud o the sum i the deomiator, S,S E[X SX S ]. We begi by igorig the fact that the sum is oly take over pairs S, S respectig P. Thus S,S E[X S X S ] = i s ( s ( s i ( s s i ( (1 p (s s i s (1 p (s ( i a i where a i := ( s i( s s i (1 p ( i ( s.

6 6 Deepak Bal et al. Here we ote the bouds ( s 4 a = Θ ad ( s 6 a 3 = Θ 3. To estimate the sum i s a i, we defie r i := a i+1 (s i = a i (i + 1( s + i + 1 (1 p i. ( Aalyzig r i will help us to aalyze a i. For example, sice r = O s < 1, we have that a > a 3. Now for x < s, defie ( 1 f(x := l r x = l(s x l(x + 1 l( s + x x l 1 p ad ote that f (x = s x 1 ( x s + x l. 1 p The first term is egative, but egligible uless x is close to s. The secod term is egative, but egligible uless x is small. The third term is always egligible as s = o(. The fourth term is positive ad costat with respect to x. Therefore, {x : f (x > 0} is a iterval. So the set {x : f(x > 0} is also a iterval ad {i : r i > 1} = {i : a i+1 > a i } is a set of cosecutive itegers. Therefore the largest term a i is either a or a i where i := 1 + max{i : r i > 1}. Also, the secod largest term is oe of a, a 3, a i, a i 1 or a i +1. To estimate i, defie i := s ( 1 1 l, ad ote that r i 1 (s i + 1 i b i ( s ( l [ ( ] s log b l 1, l 1 1 l

7 The t-toe chromatic umber of radom graphs 7 so i i. Now we will estimate a i for i i 1. First, a s = ( s 1 ( s (1 p ( s { = exp = exp s b s( s 1 [ s l s l + s 1 ]} l b { [ s (1 + o(1 l l ( l b l l b l 1 l b + 1 ( ]} l + l l b 3 l l l b l b = exp { Ω (s l l } = exp { Ω (l l l } < a 3. Now for ay i 1 i < s we have a i = ( s s i( s i (1 p ( i ( s ( s ( s s i s i ( e s i s (s i a s a s ( e s s l +1 a s {( s } exp l + 1 ( + l + l s Ω (s l l exp { Ω (l l l } < a 3. Therefore, the largest of the a i is a, ad the secod largest is a 3. I particular, ( s 4 a i a + sa 3 = O. i s Thus, applyig Jaso s iequality (3.1, we have { ( } P [X = 0] exp Ω. (3. Let B be the umber of pairs (P, U of partitios P ad sets U for which the lemma fails. We will boud E [B] usig a uio boud, liearity of expectatio, ad iequality (3.. Note that sice < U, ad sice we are lookig l for idepedet sets of size s 4 s 0 = log b log b log b 5 log b l log b ( U log b log b ( U log b l ( U

8 8 Deepak Bal et al. withi G[U], the iequality (3. applies. Now for fixed P, U, the probability that G[U] has o idepedet set of size s 0 respectig P is at most { ( } { ( } U exp Ω exp Ω s 4 l 4. Thus, E [B] s 4 0 ( { ( } 3 exp Ω k s 4 l 4. which is o(1 as log as p C 1/4 log 9/4 for C a sufficietly large costat. Here s the proof of the secod lemma: Proof (Lemma First ote that for H 1, we are doe sice G[H] ca oly have ( ( H = o H k l edges. So we tur our attetio to larger sets H. Recall the Cheroff boud: ( P[Bi(, p > (1 + δp] < exp δp for all δ >. This is a slightly modified versio of (.5 from [9]. From this we may deduce that [ ( ] ( H P E(G[H] > (1 + δ p < exp δ( H p. Settig (1 + δ ( H p = H k l δ = ad solvig for δ yields ( H 1kp l 1 = Ω ( H l So for ay fixed H with 1 < H <, the probability that G[H] has too ( ( l may edges is at most exp Ω H p = o (, so w.h.p. there are o l such sets H.. 4 Sparse graphs (p = c/ The overall pla for G = G,p with p = c/, c costat, is to first -toe color a set of vertices that icludes high degree vertices ad two eighborhoods. We will show this set is a forest ad the apply the result of [8] which says the -toe chromatic umber of a tree, T with maximum degree is τ (T = =: κ (4.1

9 The t-toe chromatic umber of radom graphs 9 The remaiig vertices will be easier to color. This process will yield a proof of Theorem. Let the vertex set of G be V, let b 0 = l 1/4 ad let For k 1, let V 0 := {v V : deg(v b 0 }. V k := V 0 k N i (V 0 (4. where N i (Z for i 1 represets the set of vertices whose distace to vertex set Z is i. Let H represet G [V ], the graph iduced o the vertex set V. I the followig proofs, we will make use of the cofiguratio model (defied below o a typical degree sequece. This is defied as follows: Defiitio A degree sequece (d 1, d,..., d is called typical if the followig three properties hold: i= i=1 d i c 3,. l 3/4 max 1 i d i l {, } 3. {i : d i b 0 } l exp l 1/4. Such degree sequeces are called typical because Lemma 3 With probability 1 o(1, the degree sequece of G is typical. Proof Property 1 follows immediately from the Cheroff iequality. 1 d i = E(G i=1 is the sum of Beroulli radom variables, which is cocetrated aroud its mea, c. It is well kow [3] that the maximum degree of G,p, p = c/ is Θ ( l l l with probability 1 o(1, so Property holds. Note that the set V0 is the same as the set o the left had side of Property 3. We have [ ( E [ V 0 ] = P Bi 1, c ] { } l 1/4 exp l 1/4 by Cheroff s iequality (see, e.g., Corollary.4 i [9]. Cosequetly, Markov s iequality yields [ { }] P V 0 > l exp l 1/4 = o(1. Lemma 4 W.h.p. H is a forest. Proof We will prove that w.h.p. H does ot have large compoets. Oce that is established, the lemma will follow from a short calculatio. We will use the followig defiitio.

10 10 Deepak Bal et al. Defiitio 3 For a graph G ad iteger i 1, let the graph G i have vertex set V (G, ad edge set E(G i = {{u, v} : d G (u, v i} Our motivatio for cosiderig G i is as follows. Suppose K is a coected compoet of H. The the set of vertices V (K V 0 iduces a coected compoet i G 5. We claim that w.h.p. G has the followig properties: P1 There does ot exist S V 0 such that S s = l 7/8 ad S iduces a coected compoet i G 5. P The maximum compoet size i H is at most l 17/8. To establish P1 ad P, fix a typical degree sequece d = (d 1, d,..., d. A radom (multi- graph with degree sequece d is costructed usig the cofiguratio model as described i Bollobás []. Let m = (d d /. We costruct a radom pairig F of the poits W = i=1 W i, W i = d i ad iterpret them as edges of a multi-graph o []. With a typical degree sequece, the probability that the resultig graph is simple is bouded away from 0 by a fuctio of c ad ot (see, e.g., [10]. We will prove that these three properties hold coditioal o a specific degree sequece, ad the sum over all degree sequeces to get the result ucoditioally. To prove P1, suppose that such a S V 0 exists. The we may assume that S = s ad that there exists a tree T i G such that the leaves of the tree are a subset of S ad V (T 5s. We may make this assumptio o V (T sice G 5 [S] is coected ad each edge i G 5 correspods to a path of legth at most 5. The P [ P1 d] is bouded above by 5s t 1 1 m i + 1 ( V 0 s t t s t t t t=s i=1 5s { } ( ( t e t l s exp s l 1/4 t t 1 c t=s 3 10 l7/8 { ( e l 5s exp s l l s l 1/4 + (5s l(5s + 10s l { } 5 l 7/8 exp l 9/8 + O(l 7/8 l l = o(1. t 1 } + O(s To see the first lie here, ote that V 0 s t s is a upper boud o the umber of ways to choose the vertices of T. t t is the umber of trees o these vertices by Cayley s formula. The umber of ways to choose cofiguratio poits correspodig to a specific tree is bouded above by ( t t sice there are at most t cofiguratio poits ad (t 1 half-edges i T. The last product is the probability that those specific cofiguratio poits are paired off i the prescribed maer.

11 The t-toe chromatic umber of radom graphs 11 To prove P, let C be a compoet of H ad let K = C V 0. The C ( K l( l 1/4 sice N(K K l( ad each of these vertices may have at most l 1/4 eighbors. But by P1, K l 7/8. So P [ P d] = o(1. Let P be the property that H is a forest. Usig these two facts we may prove P holds with high probability. We perform breadth first search to reveal H i the followig maer. We reveal the pairs of F, oe a time startig with pairs with at least oe edpoit i i: W i b 0 W i. After this, the vertices of N(V 0 have bee revealed. We the reveal pairs of F ivolvig poits correspodig to vertices of N(V 0 which reveals N (V 0. Lastly, reveal pairs where both edpoits correspod to vertices from N (V 0. At this poit H has bee revealed. Each time a edge is revealed, there is some probability that it closes a cycle. This probability is bouded above by ( l 17/8 1 m o( sice there are at most l 17/8 cofiguratio { poits correspodig } to ay particular compoet. Sice V 0 l exp l 1/4 ad l, we { } have that V l 3 exp l 1/4. There are at most V exposures total, so the uio boud gives ( P [ P d] V l 17/8 1 m o( ( 3 { } l 57/8 exp l 1/4 c = o(1. Now to remove the coditioig o d, we sum up over valid degree sequeces. P [ P] P [d ot typical] + P [ P d] P [d] = o(1 d typical by Lemma 3 ad the fact that a weighted average of o(1 terms is o(1. We ow prove Theorem by showig how to color the graph. Proof (Theorem By Lemma 4, H is a forest with probability 1 o(1. So by (4.1, we may color H with κ may colors where is the maximum degree of G. Give H such a -toe colorig ad the remove the colors o the vertices of N (V 0. This leaves a proper -toe colorig o the vertices of V 1 (recall (4.. We will ow show that the colorig o V 1 ca be greedily exteded to a proper -toe colorig of G without usig ay ew colors. Note that ay pair of vertices i V 1 at distace 1 i G receive disjoit pairs of colors. Ay pair of vertices i V 1 at distace i G receive distict pairs of colors. This was the reaso for properly colorig H ad the ucolorig

12 1 Deepak Bal et al. N (V 0. Not every proper colorig of G[V 1 ] ca be exteded to a proper colorig of V sice there may be vertices i V 1 at distace i G which are ot distace i G[V 1 ]. Let v be a ucolored vertex. We must esure that the label we assig to v is disjoit from ay curret labels o v s eighbors ad is distict from ay curret labels o vertices at distace from v. Let us cout the umber of labels that we are ot allowed to put o v. Sice v V 0, deg(v < b 0. So the umber of labels forbidde by N(v is at most b 0 κ. To see this ote that at most b 0 colors appear o vertices i N(v ad each of these colors gives rise to κ 1 labels which caot be put o v. Sice v N(V 0, N (v b 0. So the umber of labels forbidde by N (v is at most b 0, oe for each label curretly o a vertex of N (v. So we have that the umber of forbidde labels o v is at most ( b 0 κ + b κ 0 <. Hece there exists a pair of colors that we may use to label v. 5 Results for τ t (G,p, t Dese case Our mai theorem for dese radom graphs ad geeral t is a direct geeralizatio of the t = case. Theorem 3 Let p = p( satisfy C 1/4 l 9/4 p < ε < 1 where C is a sufficietly large costat ad ε is ay costat < 1. The w.h.p., τ t (G,p = (t + o(1χ(g,p. Proof (Sketch of Theorem 3 We show that w.h.p. we ca fid t partitios of [], P 1,..., P t, where each partitio cosists of (1 + o(1χ(g may idepedet sets i G, ad each partitio respects each other partitio. Oce we fid P 1,..., P t, we assig t colors to each vertex v, accordig to which part of each partitio v is i. I other words, if v P i,j P i the oe of the colors assiged to v will be c i,j. This gives a proper t-toe colorig. Ideed, sice each of the t partitios respects all the others, ay two vertices u, v share at most oe color, ad if they do share oe color the they are ot adjacet because each partitio cosists of idepedet sets. To show that the P 1... P t exist w.h.p., we use iductio o t. Suppose we are give P 1... P t 1. We will costruct P t iteratively usig Lemma ad the followig fact. Fact 1 W.h.p. for every set U [] of size U, G[U] has a idepedet set of size at least s 0 = (1 o(1α(g that respects P 1... P t 1 l.

13 The t-toe chromatic umber of radom graphs 13 Assumig this, we costruct P t by iteratively applyig Fact 1, removig idepedet sets util there are fewer tha vertices remaiig, at which l poit we apply Lemma to greedily fiish costructig the partitio P t, as was doe i Sectio 3. Proof (Sketch of Fact 1 This is aalogous to Lemma 1. Jaso s iequality gives a expoetial boud o the probability that G,p has o idepedet sets of size k respectig some fixed partitios P 1... P t 1. We let B be the umber of tuples (P 1,..., P t 1, U of partitios P i ad sets U for which Fact 1 fails. We ca the boud E [B] usig a uio boud, liearity of expectatio, ad Jaso s iequality. 5. Sparse Case Our precise result for τ (G,c/ relied o the precise result that τ (T = for ay tree T. The t-toe chromatic umber of trees is oly kow up to a costat factor. We will use the followig result of Crasto, Kim ad Kiersly: Theorem (Theorem i [6] For ay iteger t 3, there exist costats c 1, c such that for ay tree T, c 1 (T τt (T c (T. This theorem allows us to prove our result for sparse graphs: Theorem 4 Let G = G,p where p = c/ with c costat ad let t 3 be a iteger. If we let = (G represet the maximum degree, the there exist costats c 1, c such that w.h.p., c 1 τt (G,p c. Proof (Sketch of Theorem 4 The proof of this theorem is a geeralizatio of the proof of Theorem. The mai step i that proof was to prove that G[V ] is a forest. To prove this result, we will prove that H t := G[V t ] is a forest. Oe may check that the proof of Lemma 4 works i the same way for H t. For example, i property P1, we must replace G 5 with G 4t 3. For the size of the maximum compoet, we will get (l (4t+9/8. The i the calculatio for P [ P d], the expoet { of 57/8 } will be replaced by a higher costat depedig o t. However exp l 1/4 goes to zero fast eough to hadle ay polylog factor. ( Sice H t is a forest, we may t-toe color it with κ := τ t (H t = Θ may colors by Theorem. We the remove the labels except for those o

14 14 Deepak Bal et al. V t 1. This proper t-toe colorig o G[V t 1 ] may be exteded to a proper colorig of G i the same way. We took care to esure that ay two vertices of V t 1 which are at distace at most t i G receive appropriate labels. We may ow show that the remaiig vertices may be greedily colored usig o ew colors. We do this i the same way, by esurig that the maximum umber of forbidde labels at ay ucolored vertex is much smaller tha the umber of total labels. I this case, we see that the umber of forbidde labels is bouded above by t ( ( t κ b i 0 = O ( b 0 κ t 1 ( κ. i t i t i=1 Ackowledgemets Ala Frieze s research is supported i part by Natioal Sciece Foudatio Grat CCF Adrzej Dudek s research is supported i part by Simos Foudatio Grat #4471. Refereces 1. A. Bickle ad B. Phillips, t-toe Colorigs of Graphs, submitted (011.. B. Bollobás, A probabilistic proof of a asymptotic formula for the umber of labelled regular graphs, Europea Joural o Combiatorics, 1 ( B. Bollobás, Radom Graphs. Secod Editio. Cambridge Studies i Advaced Mathematics ( B. Bollobás, A. Thomaso, Set colourigs of graphs, Discrete Mathematics, 5 ( G. Chartrad, D.P. Geller, S. Hedetiemi, A geeralizatio of the chromatic umber, Mathematical Proceedigs of the Cambridge Philosophical Society, 64 (1968, D. Crasto, J. Kim, W. Kiersly, New results i t-toe colorigs of graphs, submitted (011. Preprit 7. R. Diestel, Graph Theory. Fourth Editio. Spriger ( N. Foger, J. Goss, B. Phillips, C. Segroves, Math 6450: Fial Report. homepages.wmich.edu/~zhag/fialreport.pdf 9. S. Jaso, T. Luczak, A. Ruciński, Radom Graphs. Wiley-Itersciece series i Discrete Mathematics ad Optimizatio ( B. D. McKay ad N. C. Wormald, Asymptotic eumeratio by degree sequece of graphs with degrees o(, Combiatorica, 11 (