Partitioning random graphs into monochromatic components

Transcription

1 Partitioig radom graphs ito moochromatic compoets arxiv: v4 [math.co] 15 Feb 2017 Deepak Bal Departmet of Mathematical Scieces Motclair State Uiversity Motclair, New Jersey; U.S.A. February 17, 2017 Louis DeBiasio Departmet of Mathematics Miami Uiversity Oxford, Ohio; U.S.A. Abstract Erdős, Gyárfás, ad Pyber (1991 cojectured that every r-colored complete graph ca be partitioed ito at most r 1 moochromatic compoets; this is a stregtheig of a cojecture of Lovász (1975 ad Ryser (1970 i which the compoets are oly required to form a cover. A importat partial result of Haxell ad Kohayakawa (1995 shows that a partitio ito r moochromatic compoets is possible for sufficietly large r-colored complete graphs. We start by extedig Haxell ad Kohayakawa s result to graphs with large miimum degree, the we provide some partial aalogs of their result for radom graphs. I particular, we show that if p ( 27 log 1/3, the a.a.s. i every 2- colorig of G(, p there exists a partitio ito two moochromatic compoets, 1/r, ad for r 2 if p the a.a.s. there exists a r-colorig of G(, p ( r log such that there does ot exist a cover with a bouded umber of compoets. Fially, we cosider a radom graph versio of a classic result of Gyárfás (1977 about large moochromatic compoets i r-colored complete graphs. We show that if p = ω(1, the a.a.s. i every r-colorig of G(, p there exists a moochromatic compoet of order at least (1 o(1 r 1. 1 Itroductio For a graph G ad positive iteger r, the r-color tree-partitio (tree-cover umber of G, deoted by tp r (G (tc r (G, is the miimum s such that for every r- edge-colorig of G, there exists a collectio of moochromatic coected subgraphs Research supported i part by Simos Foudatio Collaboratio Grat #

2 {H 1,..., H t } with t s such that {V (H 1,..., V (H t } forms a partitio (cover of V (G; as each subgraph H i cotais a moochromatic spaig tree we use coected subgraph, tree, ad compoet iterchageably throughout the paper. Similarly defie pp r (G, cp r (G to be the r-color path-partitio umber ad r-color cycle-partitio umber of G respectively. Gyárfás [18] oted that the followig is a equivalet formulatio of what is kow i the literature as Ryser s cojecture or the Lovász-Ryser cojecture. Cojecture 1.1 (Ryser 1970 (see [22], Lovász 1975 [28]. Let r 2. For all graphs G, tc r (G (r 1α(G. If true, this cojecture is best possible whe r 1 is a prime power by a well kow example usig affie plaes 1. For r = 2, this is equivalet to the Kőig-Egerváry theorem. Aharoi [1] proved the r = 3 case, ad for r 4 it is ope. Slightly more is kow i the case α = 1 (i.e. whe G = K, where it has bee proved for r 5 (see [19] ad [15] for more details. I a semial paper, Erdős, Gyárfás, ad Pyber [11] proved that for all r 2, ad made the followig cojecture. tp r (K pp r (K cp r (K = O(r 2 log r Cojecture 1.2 (Erdős, Gyárfás, Pyber For all r 2, tp r (K = r 1, pp r (K = cp r (K = r. For coutably ifiite complete graphs, Cojecture 1.2 is kow to be true for paths ad cycles for all r, ad kow to be true for trees whe r = 2, 3 (with the appropriate otio of paths, cycles, ad trees. Rado [33] proved pp r (K N = r; Elekes, D. Soukup, L. Soukup, ad Szetmiklóssy [10] proved cp r (K N = r; ad Nagy ad Szetmiklóssy (see [11] proved tp 3 (K N = 2. However, for fiite complete graphs the story is more complicated. For trees, a old remark of Erdős ad Rado says that a graph or its complemet is coected, i.e. tp 2 (K = 1. Erdős, Gyárfás, ad Pyber [11] proved that tp 3 (K = 2. Later, Haxell ad Kohayakawa [21] proved the followig. Theorem 1.3 (Haxell, Kohayakawa Let r 2. If 3r4 r! log r, the (1 1/r 3(r 1 tp r (K r. Later, Fujita, Furuya, Gyárfás, ad Tóth [14] cojectured that the partitio versio of Cojecture 1.1 is true ad proved it i the case whe r = 2. For paths, Gerecsér ad Gyárfás gave a simple proof of pp 2 (K = 2 (see the footote i [17]. Much later, Pokrovskiy [31] proved pp 3 (K = 3. 1 I a affie plae of order r 1, there are r parallel classes of r 1 lies each. To each of the r parallel class assig a distict color. 2

3 For cycles, Lehel cojectured that cp 2 (K = 2 (i fact, with cycles of differet colors. This was proved for large by Luczak, Rödl, ad Szemerédi [29], ad the for smaller, but still large by Alle [2], ad fially for all by Bessy ad Thomassé [5]. For geeral r, Gyárfás, Ruszikó, Sárközy, Szemerédi [20] improved the result from [11] to pp r (K cp r (K 100r log r. However for r 3, Pokrovskiy [31] proved cp r (K > r. 1.1 Large miimum degree Motivated by a ew class of Ramsey-Turá type problems raised by Schelp [35], Balogh, Barát, Gerber, Gyárfás, ad Sárközy [4] cojectured (ad proved a approximate versio of a sigificat stregtheig of Bessy ad Thomasse s result. That is, if δ(g > 3/4, the cp 2 (G 2 (with cycles of differet colors; they also provided a example which shows that the cojecture would be best possible. DeBiasio ad Nelse [8] proved that this holds for G with δ(g > (3/4 + o(1 ad the Letzter [26] proved that it holds exactly for sufficietly large. I Observatio 3.1, we ote that there are graphs with miimum degree r for which tp r (G tc r (G r ad thus it is atural to woder how small we ca make δ(g while maitaiig tp r (G r. I Theorem 4.3, we prove a stregtheig of Theorem 1.3 for graphs with large miimum degree. A corollary of our result is the followig. Corollary 1.4. For all r 2 there exists 0 such that if G is a graph o 0 vertices with δ(g > (1 1 er!, the tp r(g r. Furthermore, i Example 3.3 we show that the miimum degree i the above result caot be improved beyod (1 1. Additioally, i Theorem 4.6 we show that r+1 for coverig 2-colored graphs with two moochromatic trees, this lower boud o the miimum degree is tight. Theorem 4.3 actually gives a robust tree partitio; that is, a collectio of trees together with a liear sized set L such that after deletig ay subset of L, the remaiig graph has a tree partitio. This is importat as we will use it to obtai results o the tree partitio umber of the radom graph G(, p. Fially, as a cosequece of our method of proof, we are able to improve the boud o i Theorem 1.3. Theorem 1.5. For r 2 ad 3r 2 r! log r, tp r (K r. 1.2 Radom graphs A active area of curret research cocers sparse radom aalogs of combiatorial theorems (see the survey of Colo [7]. A early example of such a result is the so 3

4 called Radom Ramsey Theorem. Say G r H if every r-colorig of G cotais a moochromatic copy of H. For fixed graphs H, Rödl ad Ruciński [34] determied the threshold for which a.a.s. 2 G(, p r H. For the case of paths, Letzter [25] proved that for p = ω(1 a.a.s., G(, p 2 P (2/3 o(1. Radom aalogs of asymmetric Ramsey problems, hypergraph Ramsey problems, ad va der Waerde s Theorem have also bee studied (agai, see [7]. I light of these results, it is a atural questio to ask whether moochromatic partitioig problems ca be exteded to the realm of radom graphs i a iterestig way. Towards this, we prove the followig results which provide partial aalogs of Cojecture 1.1, ad Theorem 1.3 for radom graphs. Theorem 1.6. For all r 2, there exists C r such that a.a.s. (i if p ( 27 log 1/3 the tp 2 (G(, p 2, ad (ii if p ( C log 1/(r+1, the tcr (G(, p r 2, ad (iii if p ( C log 1/r, the there is a collectio of r vertex disjoit moochromatic trees which cover all but at most 9r log /p = O( 1/r (log 1 1/r vertices. Theorem 1.7. For all r 2, ( (i if p = r log ω(1 1/r, the tcr (G(, p > r, ad ( ( (ii if p = o r log 1/r, the tc r (G(, p. It is iterestig to compare Cojecture 1.1 to Theorem 1.7 as our results imply that almost every graph G G(, 1/2 (the uiform distributio o all graphs with vertices satisfies tc r (G r 2, which is much smaller tha the cojectured upper boud of (r 1α(G sice it is kow (see e.g. [13] that a.a.s. α(g 2 log 2. So ot oly are tightess examples rare, examples for which tp r (G tc r (G > r 2 are rare. 1.3 Large moochromatic compoets We cosider oe further related lie of research. Note that if a r-colored graph G ca be covered by t moochromatic compoets, the G cotais a moochromatic compoet of order at least V (G /t. So we may directly ask how large of a moochromatic compoet we may fid i a r-colored graph 3. Give a positive iteger r ad a graph G, we let tm r (G be the maximum iteger s such that the followig holds: i every r-colorig of the edges of G, there exists a moochromatic compoet with at least s vertices. For r 2, Gyarfás [18] proved tm r (K ad Füredi [16] proved 2 We say that a sequece of evets A happes a.a.s. if lim P [A ] 1. 3 Historically, the moochromatic partitioig problems metioed i the first part of the itroductio were motivated by their implicatios for graph Ramsey problems (see [32]. r 1 4

5 tm r (G for all graphs G (see Theorem 5.6 i [19]. Furthermore, this is tight (r 1α(G whe r 1 is a prime power usig the same affie plae example metioed before. Give the discussio above, ote that Füredi s result would be implied by Cojecture 1.1. Cocerig radom graphs, two sets of authors [37], [6] idepedetly foud the threshold for tm r (G(, p = Θ(. Specifically, they prove that there exists a aalytically computable costat ψ r such that if c < ψ r, the tm r (G(, c/ = o( ad if c > ψ r the tm r (G(, c/ = Ω(. We prove 4 the followig radom aalog of the fact that tm r (K. r 1 Theorem 1.8. For all r 2 ad sufficietly small ɛ > 0, there exists C such that for p C, a.a.s. every r-colorig of G(, p cotais either a moochromatic tree of order at least (1 ɛ or a moochromatic tree with at least (1 ɛ leaves, which implies r 1 tm r (G(, p (1 ɛ. r 1 Agai, it is iterestig to compare Theorem 1.8 to the correspodig determiistic versio, as our result implies that almost every graph G satisfies tm r (G (1 ɛ, r 1 which is much larger tha the boud of give by Füredi s result for which (r 1α(G there are examples showig tightess. 2 Overview ad otatio 2.1 Overview We cosider large miimum degree versios ad radom versios of some classic results for edge colored complete graphs. I certai cases we will use the large miimum degree results together with the sparse regularity lemma to obtai results for radom graphs. I these cases our approach is as follows: First, prove that edge colored graphs of high miimum degree cotai (a robust versio of the desired structure. Secod, applyig the sparse regularity lemma to the radom graph gives a reduced graph with high miimum degree ad thus we ca apply the high miimum degree result. This structure i the reduced graph correspods to a approximate spaig structure i the origial graph. As a simple applicatio of this approach we obtai Theorem 1.8. A less stadard applicatio of this approach is give i the proof of Theorem 1.6(iii where we are tryig to improve the expoet from 1/(r + 1 to 1/r. We use the method of multiple exposures to build a tree cover while maitaiig a set of vertices which are leaves i each of the moochromatic trees. O each step, the leaf set shriks by a factor of p ad at the ed of the possibly r steps, we require the leaf set to cotai more tha log vertices. By usig sparse regularity together with the large miimum degree result we are able to begi this process with a tree havig Θ( leaves as opposed to the Θ(p leaves we would be able to guaratee without sparse regularity. 4 Essetially the same result was idepedetly discovered by Dudek ad Pra lat [9]. 5

6 I Sectio 3.1, we provide examples of graphs ad colorigs which give lower bouds o tc r (G (ad hece tp r (G. I Sectio 3.2 we cosider the variat where we require that the compoets i the cover must be of distict colors. I Sectio 3.3, we give a simple upper boud o tc r (G ad prove a result about graphs i which every r + 1 vertices have a commo eighbor. Sectio 4 is devoted to provig the large miimum degree versios (icludig complete versios of our results. I Sectio 4.1 we prove Theorems 1.5 ad 4.3. The first provides a slight improvemet o the boud i Theorem 1.3 ad the secod exteds the theorem to graphs with large miimum degree. I Sectio 4.2 we prove Theorem 4.6, which provides a tight miimum degree coditio o G such that tc 2 (G 2. I Sectio 4.3, we prove that r-colored graphs with large miimum degree have large moochromatic compoets. I Sectio 5, we cotiue with the secod step of the method described above by statig the sparse regularity lemma of Kohayakawa [23] ad Rödl (see [7] as well as collectig various lemmas which will be useful for the proof. Lemma 5.7 shows that sparse edge colored radom graphs have early spaig robust tree partitios. I Sectio 6 we deduce some properties of G(, p which will be used i Sectio 7. I Sectio 7 we prove Theorem 1.6 ad 1.7. Theorem 1.6(ii ad Theorem 1.7(i,(ii will follow from the results of Sectio 6. For Theorem 1.6(i, we are able to exploit the fact that there are oly two colors to improve the geeral result of Theorem 1.6(ii. The proof of Theorem 1.6(iii is discussed i the secod paragraph of this sectio. Fially, i Sectio 7.3, we prove Theorem 1.8. I Sectio 8 we collect some cojectures ad ope problems. 2.2 Notatio We use the followig otatio throughout the paper. As usual, N(v represets the eighborhood of v ad as we deal maily with colored graphs, if c is a color the N c (v represets the eighborhood of v i the subgraph of c colored edges ad deg c (v = N c (v. If S is a set of vertices, the N c (v, S = N c (v S ad deg c (v, S = N c (v, S. We let N (S = v S N(v ad N (S = v S N(v. For two sets of vertices X ad Y, e(x, Y represets the umber of edges with oe edpoit i X ad the other i Y. For two sequeces a, b, we write a = o(b if a /b 0 as ad a = ω(b if a /b as. For costats a ad b, we write a b to mea that give b, we ca choose a small eough so that a satisfies all of ecessary coditios throughout the proof. More formally, we say that a statemet holds for a b if there is a fuctio f such that it holds for every b ad every a f(b I order to simplify the presetatio, we will ot determie these fuctios explicitly. We will igore floors ad ceiligs whe they are ot crucial to the calculatio. Logarithms are assumed to be base e uless otherwise oted. 6

7 3 Examples ad observatios 3.1 Lower bouds o the tree cover umber I this sectio we provide examples which give lower bouds o tc r (G i various settigs cosidered throughout the paper. We remid the reader that for all r 1 ad all graphs G, tp r (G tc r (G. Observatio 3.1. Let r 1. For all graphs G, if α(g r, the tc r (G r. I particular, there exists a graph G with δ(g r with tc r (G r. Proof. Choose a idepedet set {x 1,..., x r } ad color every edge icidet to x i with color i, the color the remaiig edges arbitrarily. Noe of the vertices x 1,..., x r are i a tree of the same color. Observatio 3.2. Let G = (V, E be a graph. For all 1 r s, if G cotais a idepedet set X of size s such that every vertex i V \ X has at most r 1 eighbors i X ad X is ot a domiatig set, the tc r (G > s. Proof. Start by colorig every edge i G[V \ X] with color r. The oly edges ot yet colored are those goig betwee V \ X ad X. Sice each v V \ X is icidet with at most r 1 such edges, we ca assig colors so that o vertex i V \ X is icidet with more tha oe edge of color i for ay i [r 1]. To see that G caot be covered with s moochromatic trees, ote that for ay pair x, x X, x ad x must be i differet trees; this follows sice x ad x are ot icidet with ay edges of color r ad x ad x have o eighbors of the same color (by the way colors were assiged to edges from V \ X to X, so there are o moochromatic paths from x to x. Furthermore, sice X is ot a domiatig set, there must exist at least oe tree of color r (sice every edge i G[V \ X] has color r. This implies that tc r (G s + 1. Example 3.3. For r 1 ad 2r + 2, there exists a graph G o vertices with δ(g = 1 such that tc r (G > r. r( r 1+1 r+1 Proof. There exist uique itegers m ad q such that = (r + 1m + q with 0 q r. Set aside vertices u 1,..., u r+1 ad the equitably partitio the remaiig (r + 1 vertices ito sets V 1,..., V r+1 ; that is, partitio the remaiig vertices ito sets V 1,..., V r+1 so that V 1 = = V r+1 q = m 1 ad if q 1, V r+1 q+1 = = V r+1 = m. Now add the followig colored edges: u i V j i color i for 1 i < j r + 1 u i V j i color i 1 for 1 j < i r + 1 V i V j i color r for 1 i < j r 7

8 V r+1 V i i color 1 for 2 i r V i V i with arbitrary colors for all i Note that if i j, the u i ad u j caot be i the same moochromatic compoet. Amog u 1,..., u r+1, color i oly appears icidet to u i ad u i+1 but their eighborhoods i color i are disjoit; these eighborhoods remai disjoit i color i eve whe the edges betwee the V i s are cosidered. By costructio, it is clear that u r+1 has the smallest degree of the u i ad vertices i V 1 have the smallest degree of the vertices i the V i. Note that deg(u r+1 = V V r which is r(m 1 if q = 0 ad r(m 1 + q 1 if q = 1,..., r. Now sice = (r + 1m + q, we have r( r rq = r(m r + 1 r + 1 Sice q rq+1 > q 1 for all 1 q r ad 1 r+1 r+1 = 1, ur+1 satisfies the claimed degree coditio for all 0 q r. If v 1 V 1, the deg(v 1 (r 1(m 1 + (m 2 + r = r(m 1 + r 1 deg(u r+1. u 1 u 2 u 3 u 1 u 2 u 3 u 4 u 5 V 1 V 2 V 3 V 1 V 2 V 3 V 4 V 5 Figure 1: Graphs with tc 2 (G > 2 ad tc 4 (G > 4 respectively. 3.2 Coverig with trees of distict colors Defiitio 3.4. Let G be a (multigraph. Say G has property T P r (T C r if i every r-colorig of the edges of G there is a partitio (cover of V (G with at most r trees of distict colors. Next we provide examples of graphs which caot be covered by r trees of distict colors. We ote however, that these graphs ca be partitioed ito just two compoets of the same color. Example 3.5. For all r 1 ad 2 r, there exists a graph G o vertices with δ(g = (1 1 2 r 1 such that G does ot have property T C r. 8

9 Proof. We costruct a graph G with = m 2 r vertices. The vertices are partitioed ito 2 r sets of size m. We idex these sets by biary strigs of legth r. So V = b {0,1} rv b. Every vertex withi such a set will also be referred to by the idex of its set. For a biary strig b, let b represet the strig with all bits flipped. Iclude every edge betwee vertices which agree o at least oe idex. So V b V b iff b b (all edges withi the V b are preset as well. This graph has δ(g = 1 m = ( Now color each edge with the smallest coordiate o 2 r which the edpoits idices agree. For example, two vertices with idices (0, 1, 0, 0 ad (1, 0, 0, 1, would be coected by a edge of color 3. We claim that G caot be covered by r moochromatic compoets of distict colors. Ay coected subgraph of color i ca oly cotai vertices which agree o the ith coordiate. Suppose the compoet of color i oly covers vertices with b i i the ith compoet. The the vertices with idex (b 1,..., b r are ot covered by ay of the compoets. Whe = m 2 r + q with q < 2 r, we proceed i the same way, but partitio the vertices ito q sets of size m + 1 ad 2 r q sets of size m. 3.3 Simple upper bouds o the tree cover umber Observatio 3.6. Let r 1. For all (multigraphs G, tc r (G rα(g. Proof. Let a := α(g ad let X = {x 1,..., x a } be a maximum idepedet set. So every vertex i V (G \ X has a eighbor i X. Takig the stars cetered at x 1,..., x a gives a collectio of at most rα(g moochromatic compoets which cover V (G. Propositio 3.7. Let r 2 ad let G be a graph havig the property that every set of r + 1 vertices have a commo eighbor, the tc r (G r 2. Proof. Cosider ay r-colorig of G ad let H be a r-colored auxiliary (multigraph o V (G where uv E(H of color i if ad oly if there is a path i G of color i from u to v. Sice every set of r + 1 vertices of G have a commo eighbor ad there are at most r colors, this implies α(h r. Thus by Observatio 3.6, we have tc r (H rα(h r 2. Note that a moochromatic compoet i H correspods to a moochromatic compoet i G givig the result. 4 Moochromatic trees i graphs with large miimum degree 4.1 Partitios We start by provig a lemma which we will use i the proofs of Theorems 1.5, 4.3, ad 1.6(iii. 9

10 Lemma 4.1. Let k 2. If G is a Y, Z-bipartite graph such that for all v Z, deg(v, Y > k log Z, the i every k-colorig of the edges of G, there exists a partitio {Y 1,..., Y k } of Y such that for all v Z, there exists i [k] such that N i (v Y i. Proof. Radomly color the vertices of Y with colors from [k], givig us a partitio {Y 1,..., Y k } of Y (with possibly empty parts. The probability that some vertex v Z has N i (v Y i = for all i [k] is (1 1 k deg(v,y < (1 1 k k log Z e k log Z /k = 1 Z. So by the uio boud the probability of at least oe failure is less tha 1, ad thus there exists a partitio of Y with the desired property. We ow prove Theorem 1.5 which says that tp r (K r holds provided 3r 2 r! log r (improvig the lower boud of 3r4 r! log r from Theorem 1.3 ad illustrates the idea for both Theorem 4.3 ad Theorem 1.6(iii. We ote that our proof (1 1/r 3(r 1 follows a similar procedure as the proof i [21], except that at the ed of the process we use Lemma 4.1 istead of a greedy algorithm. Proof of Theorem 1.5. Step 1: Let x 1 V (G ad let Y 1 be the largest moochromatic eighborhood of x 1, say the color is 1. Note that Y 1 ( 1/r. If every vertex i V \ Y 1 has a eighbor of color 1 i Y 1, the stop as we would already have the desired tree partitio. So some vertex x 2 has at least 1 r 1 Y 1 eighbors of say color 2 i Y 1. Set Y 2 := Y 1 N 2 (x 2. For 2 i r 1, assumig Y i has already bee defied, we do the followig: if for all v V \ Y i, ( i j=1 N j(v Y i > i log, the set k := i, Y := Y k, ad Z = V \({x 1,..., x k } Y k the proceed to Step 2. Otherwise some vertex x i+1 V \Y i has at most i log eighbors havig colors from [i] i Y i ad thus x i+1 has at least 1 r i ( Y i i log eighbors of color say i + 1, i Y i. Set Y i+1 := Y i N i+1 (x i+1. Cotiue i this maer util we go to Step 2 or util Y r has bee defied. After we complete the i = (r 1-th step, we have Y r 1 r! r 2 log j=1 r j j! r log where the last iequality holds provided ad ote that r 2 r j j=0 for 2 r 5 we directly verify r! r 2 r j log j=0 er. For r 6, we have j! log log. Recall that 3r 2 r! log r j! err! r! r 2 r j j=0, ad j! r! r 2 r j j=0. Now set k := r, Y := Y j! k, ad Z = V \ ({x 1,..., x r } Y r the proceed to Step 2. Step 2: Note that for all v Z, ( k j=1 N j(v Y k k log > k log Z. Thus we may apply Lemma 4.1 to get a partitio {Y 1,..., Y k } of Y such that for all v Z, 10

11 there exists i [k] such that N i (v Y i. Let each v Z choose a arbitrary such i ad a arbitrary eighbor i N i (v Y i. The x i alog with Y i ad all the v Z which chose eighbors i Y i form a tree of color i ad radius at most 2. Thus we have a partitio ito k r moochromatic trees. A key elemet i the precedig proof was to first build a moochromatic tree cover i which the commo itersectio of all of the trees was a large eough set of leaves. We ow explicitly defie this structure. Defiitio 4.2. A (k, l, -absorbig tree partitio is a collectio of trees T 1,..., T k together with a commo leaf set L of size l such that (i i [k] V (T i =, (ii the edges of T i have color i for all i [k], (iii every vertex i L is a leaf of T i for all i [k], ad (iv for all i j, V (T i V (T j = L. Note that if every r-colorig of a graph G o vertices cotais a (k, l, -absorbig tree partitio for some 1 k r ad l 0, the by arbitrarily assigig the leaves to the trees we have tp r (G r (with trees of distict colors. We will cosider absorbig tree partitios i two differet settigs: first, i Theorem 4.3 we wish to optimize the boud o the miimum degree so that tp r (G r, ad secod, we will apply Theorem 4.3 i a settig where the graph is early complete, i which case we do ot eed so much cotrol over the miimum degree as we eed cotrol over the size of the commo leaf set. So for the purposes of streamliig, we combie everythig we wat ito the followig statemet, which has a parameter ɛ related to the miimum degree ad a parameter α which is related to the size of the leaf set ad the lower boud o. The method of proof will be similar to that of Theorem 1.5; however, the calculatios are differet as here we are attemptig to optimize the miimum degree istead of the lower boud o. Theorem 4.3. Let r 2, 0 < ɛ < 1, α = 1 ɛ, ad er! er! 0 = max{ 12 log( 6, 4r 2r log( }. α 2 α 2 α α If G is a graph o 0 vertices with δ(g (1 ɛ, the i every r-colorig of G there either exists a (1, l, -absorbig tree partitio with l 2 log (i.e. a α moochromatic spaig tree with at least 2 log leaves or a (k, l, -absorbig α tree partitio with 2 k r ad l α/2. We will eed the followig two statemets i the proof of Theorem 4.3. Observatio 4.4. Let x R with x 2. If 2x log x, the log < 1 x. Proof. We first ote that log is strictly decreasig sice 2x log x > e. Now sice 2x log x < x 2 log(2x log x log(2x log x, we have = < 1. 2x log x x log x 2 x 11

12 Lemma 4.5. Let 0 < α 1 ad 0 = 12 log( 6 ad let G be a graph o α 2 α 2 0 vertices. If there exists x V (G such that for all v V (G, deg(v, N(x α, the G has a spaig tree with at least 2 log leaves. α Proof. We will show that x alog with at most 2 log of its eighbors form a domiatig α set. Set Y 1 := N(x ad Z 1 = V \({x} Y 1. For 1 i 3 log 1, do the followig: 2α If Z i, let y i be the vertex i Y i with the largest degree to Z i. Set Y i+1 = Y i \{y i } ad Z i+1 = Z i \ N(y i. Sice deg(y i, Z i α i Z Y 1 i i > 2α Z 3 i (where the secod iequality holds by Observatio 4.4 ad the boud o ad thus Z i+1 < (1 2α/3 Z i (1 2α/3 i Z 1 (1 2α/3 i+1 1 whe i log. Thus whe the process stops, we have a spaig tree with at 2α most 3 log 2 log o-leaves. 2α α Proof of Theorem 4.3. Step 1: Let x 1 V (G ad let Y 1 be the largest moochromatic eighborhood of x 1, say the color is 1. Note that Y 1 1 (1 ɛ. If for all v r V \ Y 1, deg 1 (v, Y 1 α, the sice 12 log( 6, we may apply Lemma 4.5 to get α 2 α 2 a moochromatic spaig tree (i color 1 with at least 2 log leaves ad we are α 1 doe. Otherwise some vertex x 2 has at least ( Y r 1 1 ɛ α eighbors of say color 2 i Y 1. Set Y 2 := Y 1 N 2 (x 2. For i 2, do the followig: if for all v V \ Y i, ( i j=1 N j(v Y i α, the set k := i, Y := Y k, ad Z = V \ ({x 1,..., x k } Y k the proceed to Step 2. Otherwise 1 some vertex x i+1 V \ Y i has at least ( Y r i i ɛ α eighbors of color say i + 1, i Y i. Set Y i+1 := Y i N i+1 (x i+1. Cotiue i this maer, util we go to Step 2 or util i = r. If i = r, the Y r ( 1 r r! ɛ j=1 r 1 1 j! α j=1 ( 1 1 ɛ(e 1 α(e 1 j! r! ad thus every vertex i V \ Y r has at least ( 1 Y r ɛ ɛ(e 1 α(e 1 ɛ = α r! eighbors i Y r. Now set k := r, Y := Y k, ad Z = V \ ({x 1,..., x k } Y k the proceed to Step 2. Step 2: First set aside α/2 vertices from Y to be the commo leaf set of the absorbig tree partitio ad let Y be the remaiig vertices i Y. Every vertex i Z still has at least α/2 eighbors i Y. Sice 4r 2r log(, Observatio 4.4 implies that α α α/2 > r log ad thus we ca apply Lemma 4.1 to get a partitio {Y 1,..., Y k } of Y such that for all v Z, there exists i [k] such that N i (v Y i. By arbitrarily choosig such a Y i for each v Z, we have a (k, l, -absorbig tree partitio with 2 k r ad l α/2. 12

13 While we are ot able to prove that the boud o the miimum degree i Theorem 4.3 is optimal, Observatio 3.1 shows that there are graphs with δ(g r for which tp r (G tc r (G r, ad thus the goal i the miimum degree versio of the problem (optimizig δ(g while maitaiig tp r (G r is differet from the goal i the case of complete graphs (provig tp r (K r 1. I Theorem 4.3 we actually prove that the trees have distict colors, so it is atural to ask the questio of how the miimum degree threshold for partitioig (coverig ito r trees compares to the miimum degree threshold for partitioig (coverig ito r trees of distict colors (see Sectio Covers For the cover versio of the r = 2 case, we ca actually prove a tight boud o the miimum degree (see Example 3.3. Theorem 4.6. Let 1. tc 2 (G 2. For all graphs G o vertices, if δ(g 2 5 3, the Proof. Suppose that = 3m + q where q {0, 1, 2}. The δ(g 2 5 traslates to 3 δ(g 2m 1 + q 2. Suppose G is 2-colored ad let T = {R 1,..., R k, B 1,..., B l } be a moochromatic compoet cover of G with the fewest umber of compoets, where each compoet is maximal; ad with respect to this, choose T so that as may differet colors are represeted as possible. Without loss of geerality suppose k l. We are doe uless T 3. It is clear from miimality of the umber of compoets, that for each compoet T T, there is a o-empty subset of vertices φ(t such that every vertex i φ(t is ot cotaied i ay other compoet S T \ {T }. Case 1 (There is at least oe compoet of each color. Sice k l ad k + l 3, we have k 2. Suppose first that there exist vertices u i φ(r i, u j φ(r j, ad v h φ(b h such that u i u j E(G. By the maximality of the compoets, u i v h, u j v h E(G. So N(u i N(u j N(v h 3 3( 3 δ(g = 3δ(G So let w N(u i N(u j N(v h. If w is i a blue compoet B T, the w caot be adjacet to u i or u j via a blue edge (as this would imply that u i or u j is cotaied i B. So w is adjacet to u i ad u j via red edges, but this cotradicts the fact that u i ad u j are i differet red compoets. So suppose that for all blue compoets B T, w V (B. This implies that wv h must be a red edge ad that w is i a red compoet of T, but agai this cotradicts the fact that v h is ot cotaied i ay red compoet of T. I either case, we get a cotradictio. So we may assume that the vertices of φ(r 1,..., φ(r k iduce a complete k-partite blue graph B. However, this implies that {B 1,..., B l, B } is a cover with fewer compoets. 13

14 Case 2 (All of the compoets have the same color. Sice k l, T = {R 1,..., R k }. Without loss of geerality suppose R 1 R 2 R k. Note that sice k 3 ad all compoets are red, we have 2 R 1 m. Sice R 1 is maximal, every edge leavig R 1 is blue ad thus for all v R 1 we have q q N B (v (V (G \ R 1 δ(g ( R 1 1 2m + R 1 m +. (1 2 2 From (1, ad the fact that V (G \ R 1 3m + q 2, we see that for ay set of three vertices {x, y, z} i R 1, some pair of {x, y, z} must have a commo blue eighbor i V (G \ R 1. This implies that there are either oe or two blue compoets which cover the vertices of R 1. If there were oly oe, we would be i Case 1. So assume that there are two blue compoets B 1 ad B 2 which cover every vertex i R 1. Now usig (1, we get that q q q B 1 + B 2 R 1 +2(2m+ R 1 = 4m+2 R 1 3m+2 3m+q So either B 1 ad B 2 form a cover with two compoets, or B 1, B 2, ad the red compoet cotaiig the leftover vertex form a cover with two blue compoets ad a red compoet ad thus we are i Case Large moochromatic compoets We use the followig lemma of Liu, Morris, ad Price [27] (a essetially equivalet versio of this lemma was idepedetly proved by Mubayi [30]. A double-star is a tree havig at most two vertices which are ot leaves. Lemma 4.7 (Lemma 9 i [27]. Let c 0 ad let G be a X, Y -bipartite graph o vertices. If e(g c X Y, the G has a double-star of order at least c. Theorem 4.8. Let r 2, ad let 0 < ɛ 1. If G is a graph o vertices with 2 δ(g (1 ɛ, the every r-colorig of G cotais either a moochromatic tree of order at least (1 ɛ or a moochromatic double-star of order at least (1 2ɛ. r 1 Proof. Let H be the largest moochromatic tree i G, say of color r ad suppose that V (H < (1 ɛ. Set X = V (H ad Y = V (G \ X ad let B = G[X, Y ] be the bipartite graph iduced by the bipartitio {X, Y }. Without loss of geerality suppose X Y ad ote that e(b Y ( X ɛ (1 2ɛ X Y. Note that by the maximality of H, there are o edges of color r i B, so at least (1 2ɛ 1 X Y of the r 1 edges of B are say color 1. So by Lemma 4.7, we have a moochromatic double-star with at least (1 2ɛ vertices. r 1 14

15 5 Sparse Regularity ad Basic Applicatios We will use of a variat of Szemerédi s regularity lemma [38] for sparse graphs, which was proved idepedetly by Kohayakawa [23] ad Rödl (see [7] ad later geeralized by Scott [36]. Say that a U, V -bipartite graph is weakly-(ɛ, q-regular if for all U U ad V V with U ɛ U ad V ɛ V, e(u, V q. Give disjoit sets X, Y ad 0 < p 1, defie the p-desity of (X, Y, deoted d p (X, Y, by d p (X, Y = e(x, Y p X Y. We say the bipartite graph G[U, V ] iduced by disjoit sets U ad V is (ɛ, p-regular or that (U, V is a (ɛ, p-regular pair if for all subsets U U, V V with U ɛ U, V ɛ V we have d p (U, V d p (U, V ɛ. Give a graph G = (V, E, a partitio {V 1,..., V k } of V is said to be a (ɛ, p-regular partitio if it is a equitable partitio (i.e. V i V j 1 for all i, j [k] ad all but at most ɛ ( k 2 pairs (Vi, V j iduce (ɛ, p-regular pairs. We will use the followig r-colored versio of the sparse regularity lemma due to Scott [36, Theorem 4.1]. Lemma 5.1. For every ɛ > 0 ad m, r 1, there exists M such that if G 1,..., G r are edge-disjoit graphs o vertex set V with V m ad where p i = e(g i > 0 is the ( V 2 desity of G i for all i [r], there exists a partitio {V 1,..., V k } of V with m k M such that for all i [r], {V 1,..., V k } is a (ɛ, p i -regular partitio of G i. Let G be a r-colored graph o vertices with p = e(g ( 2 ad r-colorig {G 1,..., G r }, where for all i [r], p i = e(g i ( 2 ; ote that p = r i=1 p i. We defie the (ɛ, p, δ-reduced graph Γ as follows: Let {V 1,..., V k } be the partitio of G obtaied from a applicatio of Lemma 5.1. Let V (Γ = {V 1,..., V k } ad say that {V i, V j } is a c-colored edge of Γ if the p c -desity of (V i, V j i G c is at least δ. Note that Γ is a (possibly multicolored graph. The followig simple lemma is typically applied to the reduced graph obtaied after a applicatio of Lemma 5.1. Lemma 5.2. Let α > 0. If H is a graph o k 2 α vertices with e(h (1 α ( k 2, the there exists a subgraph H H such that V (H (1 αk ad δ(h (1 2 αk. α k 2 Proof. As there are at most α ( k o-edges, there are at most αk vertices V 2 2 which have at least α k o-eighbors. Let 2 H = H[V ]. We have V (H k αk = (1 αk ad δ(h k 1 α k αk (1 2 αk. 2 15

16 We will apply Lemma 5.1 to a r-colored graph G G(, p with p = ω(1 to get a partitio {V 1,..., V k }. We will oly require the very weak coditio that each pair which is (ɛ, p i -regular for all i [r] is weakly-(ɛ, 1-regular i some color c [r]. This will follow as a simple cosequece of the uio boud ad the Cheroff boud, which we state first for coveiece (see e.g. Corollary 21.7 i [13]. Theorem 5.3 (Cheroff. Let X be a biomially distributed radom variable. The for α > 0, P [X (1 αe [X]] exp( α 2 E [X] /2. Lemma 5.4. For all η > 0, there exists C such that if p C, the a.a.s. G G(, p has the property that for all X, Y V (G with X Y = ad X, Y η, 3p X Y /2 e(x, Y p X Y /2. Lemma 5.5. Suppose we have applied Lemma 5.1 with 0 < ɛ < 1 to a r-colored graph 2r G G(, p with p = ω(1 to get a partitio {V 1,..., V k }. The every pair (V i, V j which is (ɛ, p l -regular for all l [r] is weakly-(ɛ, 1-regular i some color c [r]. Proof. Suppose the pair (V i, V j is (ɛ, p l -regular for all l [r]. Lemma 5.4 implies that e(v i, V j p V i V j /2 ad thus for some color c [r], we have e c (V i, V j p V i V j /2r. So if V i V i ad V j V j with V i, V j ɛ, the by (ɛ, p k c-regularity, we have ad so e c (V i, V j p c V i V j = d p c (V i, V j d pc (V i, V j ɛ = e c(v i, V j p c V i V j ɛ e c (V i, V j ( p 2r ɛp c V i V j > 0, ad thus (V i, V j is weakly-(ɛ, 1-regular i color c. 5.1 Nearly spaig tree partitios p 2rp c ɛ Lemma 5.6. Let 0 < ɛ < 1/3 ad let G be a U, V -bipartite graph. If G is weakly (ɛ, 1-regular, the G cotais a tree T with leaf set L such that (i V (T U (1 ɛ U, V (T V (1 ɛ V, ad (ii L U (1 4ɛ U, L V (1 4ɛ V. Proof. Let U be a subset of U of size exactly 3ɛ U ad let V be a subset of V of size exactly 3ɛ V. Let G be the graph iduced by U, V ad suppose first that o compoet of G itersects U i at least ɛ U vertices or itersects V i at least ɛ V vertices. However, ow we may partitio U ito sets U 1 ad U 2 ad V ito sets V 1 ad V 2 such that ɛ U U 1 2ɛ U, ɛ V V 1 2ɛ V, ad there are o edges from 16

17 U 1 V 1 to U 2 V 2 ; however, this cotradicts the fact that G is weakly (ɛ, 1-regular. So suppose that some compoet H of G itersects say U i at least ɛ U vertices. The sice G is weakly-(ɛ, 1-regular, we see that all but at most ɛ V vertices of V are also i H ad thus there exists a coected subgraph H of G which itersects each of U ad V i at most 3ɛ U ad 3ɛ V vertices respectively. Fially, usig the fact that G is weakly (ɛ, 1-regular, we see that all but at most ɛ U vertices of U ad ɛ V vertices of V have a eighbor i H, ad thus G cotais a tree i which at least (1 4ɛ U vertices of U ad at least (1 4ɛ V vertices of V are leaves. Say that a graph G o vertices has property T (r, l, λ if every r-colorig of G either has a moochromatic tree with at least (1 λ leaves or G has a (s, l, -absorbig tree partitio for some 2 s r ad (1 λ. Lemma 5.7. Let r 1 ad 0 < ɛ < 1 T (r,, 6ɛ. 8er! 10r ω(1. If p =, the a.a.s. G(, p has property Proof. Let G G(, p be r-colored. Let m be large eough so that log m < ɛ ( 1 ɛ m 4r er! ad let ɛ := ɛ2. Apply Lemma 5.1 to G with 4r ɛ, m, ad r. Let Γ be the (ɛ, p, 1/2r- reduced graph obtaied. By Lemma 5.2, we ca pass to a subgraph Γ Γ with k := Γ (1 rɛ k = (1 ɛ/2k ad δ(γ (1 2 rɛ k (1 ɛk. We color the edges of Γ by a arbitrary color c guarateed by Lemma 5.5 ad recall that this says the c-colored edges of Γ represet c-colored weakly-(ɛ, 1-regular pairs i G. Now apply Theorem 4.3 with ɛ to Γ to get a (s, l, k -absorbig tree partitio T with l k 2 log α k (1 ɛk (where the secod iequality holds by the choice of m ad sice k m if s = 1 ad l k /(4er! if 2 s r. Sice the edges of Γ represet weakly-(ɛ, 1-regular pairs, we ca apply Lemma 5.6 to each edge of T to see that G a.a.s. has property T (r,, 6ɛ; that is, to get the absorbig tree partitio T of 8er! G. Note that if there are τk leaves i T, each of which is a leaf i s differet trees, we will get (by Lemma 5.6(ii at least τk (1 4sɛ (1 5sɛτ leaves i the origial k graph which are leaves i all s trees. So the total umber of leaves i T is at least { (1 5sɛ(1 ɛ (1 6ɛ if s = 1 (1 5sɛτ (1 5sɛ 1 4er! 8er! if 2 s r Also ote that sice T covers the k vertices of Γ, T will cover (by Lemma 5.6(i at least k (1 ɛ (1 ɛ/2k (1 ɛ (1 2ɛ vertices of G as desired. k k 17

18 6 Lemmas for Radom Graphs The followig lemma follows from stadard applicatios of the Cheroff ad uio bouds. Lemma 6.1. Let r 1. If p ( 9r log 1/r, the a.a.s., every ( set R of r vertices i ( G(, p satisfies N (R p r /2. Furthermore, if p = ω log 1/r, the for ay ɛ > 0, a.a.s., every set R of r vertices satisfies (1 ɛp r N (R (1 + ɛp r. Erdős, Palmer, ad Robiso [12] determied the exact threshold for whe the eighborhood of every vertex (of degree at least 2 i G(, p iduces a coected subgraph. We eed the followig lemma which gives us a boud o the value of p for which the commo eighborhood of every set of r vertices i G(, p is o-empty ad iduces a coected graph. Lemma 6.2. Let r 1. If p ( C log 1/(r+1 with C sufficietly large, the a.a.s. i G G(, p, G[N (R] is coected for every set R of r vertices. Proof. Let 0 < ɛ 1/2, let 3(r < C < C1/(r+1 2, ad suppose (1 ɛp r m (1 + ɛp r. The C log m m C log (1 ɛp r C (1 ɛ ( 1/(r+1 ( 1/(r+1 log C log < p. (2 Usig (2, the probability that the subgraph of G iduced by a set of size m is discoected is at most m/2 k=1 ( m/2 m (1 p k(m k k k=1 k=1 ( exp k log ( me k(m k C log m k m m/2 ( ( exp k (log m C 1 2 log m = o. r+1 There are O( r r-sets R for which we must cosider N (R (which by Lemma 6.1 satisfy (1 ɛp r N (R (1 + ɛp r, ad thus the expected umber of N(R which iduce a discoected subgraph teds to 0. Lemma 6.3. Cosider G G(, p with vertex set V. The a.a.s. for ay set L V 80 log with L, all but at most 9 log of v V \ L satisfy N(v, L L p/2. p p 18

19 Proof. For a fixed set L, we have that N(v, L Bi( L, p, so the Cheroff Boud implies that P [ N(v, L < L p/2] e L p 8. Call v bad for L if N(v, L < L p/2. Sice N(v, L ad N(u, L are idepedet for differet vertices v ad u, we have that the probability that there exists a L with at least 9 log /p may bad vertices is at most l=80 log /p ( ( l 9 log /p ( exp lp 8 9 log ( exp l log 9(log 2 + p 8 p l=80 log /p (log 2 exp (log = o(1 p The followig Lemma will be used to prove Theorem 1.7. We prove it here as it may be of idepedet iterest. Lemma 6.4. ( Let r 1. 1/r (i If p = r log ω(1 ad G G(, p, the a.a.s. there exists a set S V (G such that S = r, S is idepedet, S is ot a domiatig set, ad for all v V (G, deg(v, S r 1. (ii For all s > r with s log s = o(log ad 0 < c < 2( 1 r, if p ( c log 1/r s ad G G(, p, the a.a.s. there exists a set S V (G such that S = s, S is idepedet, S is ot a domiatig set, ad for all v V (G, deg(v, S r 1. Proof. We begi with the proof of (ii. Choose s, c, ad p as i the statemet. We first show that we ca fid a idepedet set S of s vertices such that o r vertices i S have a commo eighbor (or equivaletly, o v V \ S has more tha r 1 eighbors i S. For S ( [] s let XS be the idicator radom variable for the evet ad let A S := {S is idepedet ad o r vertices i S have a commo eighbor} k=r X = S ( [] s For a fixed set S, let q represet the probability that a vertex v S has at least r eighbors i S. The s ( ( ( ( s r s s ( s q := p k (1 p s k = p r (1 p s r + O (sp t = p r (1+O(sp. k r r 19 X S. t=1

20 To see the last two equalities, ote that the coditios o s imply that sp = o(1 ad for all t > 0, ( s r+t p r+t (sp t. p r ( s r Now we have that P [X S = 1] = (1 q s (1 p (s 2 ad E [X] = ( ( (1 q s (1 p (s2 = exp s log s ( e s ( s c log (1 + o(1 + O(1 r sice ( s r c < 1 < s ad s log s = o(log. A applicatio of the secod momet method will show that such a set exists a.a.s. It suffices to show that E [X 2 ] /E [X] o(1 (see e.g. Corollary i [3]. E [ X 2] = E = E [X] + S ( [] s s 1 k=0 X S 2 S 1 S 2 =k P [A S1 A S2 ] (3 ( 2 s 1 E [X] + (1 q 2 4s (1 p 2(s 2 + O ( 2s k. s The secod sum i (3 is over ordered pairs of sets. Sice E [X] 2 = ( 2(1 ( s q 2 2s (1 ( p 2(s 2 = Ω 2s s e 2( rc s log, we have E [ ( X 2] /E [X] 2 1 E [X] + (1 s 2s+1 e 2(s rc log q 2s + O 1 + o(1 sice s log s = o(log ad 2 ( s r c < 1. Now ote that a.a.s., N (S = O(sp = o( ad so S is ot a domiatig set ad (ii is proved. The proof of (i is similar, but we are more careful i the calculatio of E [X 2 ]. Set ω := ω(1 ad suppose p = ( r log ω 1/r. We wat to prove the existece of a set S of size r. I this case we simply have q = p r, ad thus P [X S = 1] = (1 p r r (1 p (r 2 k=1 20

21 ad ( E [X] = (1 p r r (1 p (r 2 r ( = exp r log r log ω + O(1 = exp (ω(1 + o(1. For r-sets R 1, R 2 with R 1 R 2 = k, we have Thus P [A R1 A R2 ] ( p k (1 p r k 2 + (1 p k 2r (1 p ( r 2+k(r k+( r k 2 E [ X 2] = E [X] + ( p k (1 p r k 2 + (1 p k 2r r 1 k=0 R 1 R 2 =k P [A R1 A R2 ] r 1 ( ( ( r r (p E [X] + k (1 p r k 2 + (1 p k 2r r k r k k=0 ( 2 r 1 E [X] + (1 p r 2 4r + exp (2ω k log + O(1. r Sice E [X] 2 = ( r 2(1 p r 2 2r (1 p 2(r 2 = exp (2ω(1 + o(1, we agai have that k=1 E [ X 2] /E [X] o(1. 7 Moochromatic trees i radom graphs 7.1 Upper bouds o the tree cover/partitio umber Theorem 7.1. For all r 2, there exists C r such that a.a.s. (i if p ( 27 log 1/3 the tp 2 (G(, p 2, ad (ii if p ( C log 1/(r+1, the tcr (G(, p r 2, ad (iii if p ( C log 1/r, the there is a collectio of r vertex disjoit moochromatic trees which cover all but at most 9r log /p vertices. 21

22 Proof. Part (ii follows directly from Propositio 3.7 ad Lemma 6.1. For part (i, suppose the edges have bee colored with colors 1 ad 2 ad cosider two vertices u ad v with o moochromatic path betwee them; if there were o such pair, we would have a spaig moochromatic compoet by the remark of Erdős ad Rado. By Lemma 6.1, N ({u, v} p 2 /2. Let N x,c with x {u, v} ad c {1, 2} represet the color c eighbors of vertex x i N ({u, v}. The we must have that N u,1 N v,1 = N u,2 N v,2 =. Thus N u,1 = N v,2 =: A ad N u,2 = N v,1 =: B. Now by Lemma 6.2, N ({u, v} iduces a coected subgraph. If both A ad B are oempty the there is a edge betwee them, but this edge would give a moochromatic path betwee u ad v. Thus wlog, N ({u, v} = A. By Lemma 6.1, every vertex i Z := V N ({u, v} {u, v} has at least p 3 /2 27 log /2 may eighbors i N ({u, v}. So applyig Lemma 4.1, with k = 2, Y = N ({u, v}, ad Z as above, we have obtaied the desired partitio ito two moochromatic trees. I order to prove part (iii, we will use the method of multiple exposures. Let p ( C log 1/r with C > 2000er!r(4r 2 r ad let p be such that (1 p r+1 = (1 p. Note that i this case, p p p. The we may view G(, p as G r+1 0 G r where each G i G(, p for 0 i r. We will expose the G i oe at a time alog with the colors assiged to their edges. Note that if a edge belogs to more tha oe G i, the whe it is revealed a secod time, we already kow its color. This does ot affect our argumet. Let α = 1. First we expose G 8er! 0 ad apply Lemma 5.7 with ɛ as ay small costat. This provides us with a (s, α, -absorbig tree partitio T ad commo leaf set L 0, such that 1 s r ad (1 6ɛ. If =, the we are doe. Our goal is ow to attach as may of the vertices of V 0 := V (G \ V (T to L 0 as possible. Expose G 1 (ad all the colors assiged to its edges. Apply Lemma 6.3 to G 1 with L 0 as L. This is possible sice L 0 p α p 80 log. Let V 0 = {v V 0 : N(v, L 0 L 0 p/2} ad let V 0 = V 0 \ V 0. The by the lemma, V 0 9 log /p. Now if every vertex i V 0 has at least r log eighbors i L 0 with colors from [s] (ote that if s = r, the this must be the case, the we may apply Lemma 4.1 with L 0 as Y ad V 0 as Z to get a partitio {Y 1,..., Y s } of L 0 such that for all v V 0 there exists l [s] such that N l (v Y l. By arbitrarily choosig such a Y l for each v V 0, we have the desired tree partitio of V \ V 0. Otherwise there is a vertex x 1 i V 0 satisfyig N j (x 1, L 0 ( L 0 p 2 r log /(r s α p 2r for some j [r] \ [s]. Without loss of geerality, j = s + 1. Set L 1 := N s+1 (x 1, L 0 ad V 1 = V 0 {x 1 }. Now suppose that for some 1 i r s, we have foud vertices {x 1,..., x i } V ( 0 p i, ad sets L j for 1 j i such that L i α 2r Li L i 1 ad L i N j (x j for 1 j i. Expose G i+1 ad apply Lemma 6.3 to G i+1 with L i as L. We may apply the 22

23 lemma sice ( i ( r p p L i p α p α 80r log. 2r 2r(r + 1 Let V i = {v V i : N(v, L i L i p/2} ad let V i = V i \ V i. If every vertex i V i has at least r log eighbors i L i with colors from [s + i] (if s + i = r, the this is the case by the calculatio above, the we may apply Lemma 4.1 ad we are doe as i the base case. Otherwise, there is a vertex x i+1 V i satisfyig N j (x i+1, L i ( L i p 2 ( i+1 p r log /(r (s + i α 2r for some j [r] \ [s + i]. Without loss of geerality, j = s + i + 1. Set L i+1 := N s+i+1 (x i+1, L i ad V i+1 = V i {x i+1 }. Thus we will fid the desired partitio after at most r s iteratios of the above procedure. At each stage we lose at most 9 log /p may vertices ad thus we lose at most 9r log /p i total. 7.2 Lower bouds o the tree cover umber Theorem 7.2. For all r 2, ( (i if p = r log ω(1 1/r, the tcr (G(, p > r, ad ( ( (ii if p = o r log 1/r, the tc r (G(, p. Proof. (i We apply Lemma 6.4 (i to get a idepedet set of size r which is ot domiatig ad with o commo eighbor. But the Observatio 3.2 applied with s = r fiishes the proof. (ii Choose s as a fuctio of so that s, but s log s = o(log. Similarly to the previous part, the proof follows by applyig Lemma 6.4 (ii ad Observatio Large moochromatic compoets Fially, we prove that for all r 2 ad 0 < ɛ 1/r, there exists C > 0 such that if p C, the a.a.s. tm r(g(, p (1 ɛ r 1. Proof of Theorem 1.8. Let r 2, 0 < ɛ < mi{1/2r, 1/9}, m 1/ɛ 2, ad let M be give by Lemma 5.1. Choose C sufficietly large for a applicatio of Lemma 5.4, let p C, ad let G 1,..., G r be a edge colorig of G G(, p. Apply Lemma 5.1 with ɛ 4 /r 2, m, ad r ad the Lemma 5.2 to get a cleaed-up reduced graph Γ o k (1 ɛ 2 k vertices with δ(γ (1 2ɛ 2 k. Apply Theorem 4.8 to Γ to get a moochromatic tree T i Γ with either T (1 2ɛ 2 k or T havig at 23

24 least (1 4ɛ 2 k /(r 1 leaves. Apply Lemma 5.6 to each edge of T to get the desired tree T. I the secod case, ote that the umber of leaves of T is at least (1 4ɛ 2 k r 1 (1 4ɛ 2 k (1 4ɛ2 2 (1 ɛ 2 r 1 (1 ɛ r 1, where the last iequality holds sice ɛ 1/9. 8 Ope Problems The mai ope problem is to improve Theorem 1.6.(iii so as to avoid the eed for the leftover vertices. We make the followig cojecture ad ote that for r 3 it would be iterestig to get ay o-trivial boud o p such that tp r (G(, p r. Cojecture 8.1. For all ɛ > 0 ad r 1, if p r. ( (1+ɛr log 1/r, the a.a.s. tpr (G(, p Note: While this paper was uder review, Kohayakawa, Mota, ad Schacht [24] proved the r = 2 case of Cojecture 8.1. It would be iterestig to exted Theorem 4.6 to a partitio versio. Cojecture 8.2. For all graphs G o vertices, if δ(g 2 5 3, the tp 2 (G 2. For the followig cojecture, Theorem 4.6 provides the r = 2 case, ad for the r = 1 case, ote that if δ(g 1, the G is coected, i.e. tp 2 1(G = tc 1 (G = 1. Furthermore, Example 3.3 shows that this is best possible if true. Cojecture 8.3. For all r 1, if G is a graph o vertices with δ(g r( r 1+1 r+1, the tc r (G r. Regardig the distict colors variat of these problems, we make the followig cojecture which is true for r = 1 as above, ad for r = 2 by Letzter s result [26]. Furthermore, Example 3.5 shows that this is best possible if true. Cojecture 8.4. Let r 1. If δ(g (1 1 2 r, the G has property T C r (T P r. Fially, i Theorem 4.3 we prove that if a r-colored graph G has sufficietly large miimum degree, the G ca be partitioed ito r moochromatic trees, each of which implicitly has may leaves. What about partitioig ito trees with few leaves? Problem 8.5. For all r 2, sufficietly small ɛ > 0, ad sufficietly large 0, if G is a graph o 0 vertices with δ(g (1 ɛ, the i every r-colorig of G there exists a partitio of G ito O(r moochromatic trees so that each tree has O(1 leaves. 24

25 9 Ackowledgemets We thak Rajko Neadov, Frak Mousset, Nemaja Škorić ad idepedetly Hiệp Há for drawig our attetio to a error i a earlier versio of this paper related to Theorem 1.6. We also thak the referees for makig may useful commets which helped us improve the orgaizatio of the paper. Refereces [1] R. Aharoi. Ryser s cojecture for tripartite 3-graphs. Combiatorica, 21(1:1 4, [2] P. Alle. Coverig two-edge-coloured complete graphs with two disjoit moochromatic cycles. Combiatorics, Probability ad Computig, 17(04: , [3] N. Alo ad J. H. Specer. The probabilistic method. Wiley-Itersciece Series i Discrete Mathematics ad Optimizatio. Joh Wiley & Sos, Ic., Hoboke, NJ, third editio, With a appedix o the life ad work of Paul Erdős. [4] J. Balogh, J. Barát, D. Gerber, A. Gyárfás, ad G. N. Sárközy. Partitioig 2-edge-colored graphs by moochromatic paths ad cycles. Combiatorica, 34(5: , [5] S. Bessy ad S. Thomassé. Partitioig a graph ito a cycle ad a aticycle, a proof of lehel s cojecture. Joural of Combiatorial Theory, Series B, 100(2: , [6] T. Bohma, A. Frieze, M. Krivelevich, P. Loh, ad B. Sudakov. Ramsey games with giats. Radom Structures Algorithms, 38(1-2:1 32, [7] D. Colo. Combiatorial theorems relative to a radom set. arxiv preprit arxiv: , [8] L. DeBiasio ad L. Nelse. Moochromatic cycle partitios of graphs with large miimum degree. Joural of Combiatorial Theory, Series B, 122: , [9] A. Dudek ad P. Pra lat. O some multicolour ramsey properties of radom graphs. arxiv preprit arxiv: , [10] M. Elekes, D. T Soukup, L. Soukup, ad Z. Szetmiklóssy. Decompositios of edge-colored ifiite complete graphs ito moochromatic paths. arxiv preprit arxiv: , [11] P. Erdős, A. Gyárfás, ad L. Pyber. Vertex coverigs by moochromatic cycles ad trees. Joural of Combiatorial Theory, Series B, 51(1:90 95,