Rainbow Hamilton cycles in random graphs

Transcription

1 Raibow Hamilto cycles i radom graphs Ala Frieze Po-She Loh Abstract Oe of the most famous results i the theory of radom graphs establishes that the threshold for Hamiltoicity i the Erdős-Réyi radom graph G,p is aroud p log+loglog. Much research has bee doe to exted this to icreasigly challegig radom structures. I particular, a recet result by Frieze determied the asymptotic threshold for a loose Hamilto cycle i the radom 3-uiform hypergraph by coectig 3-uiform hypergraphs to edge-colored graphs. I this work, we cosider that settig of edge-colored graphs, ad prove a result which achieves the best possible first order costat. Specifically, whe the edges of G,p are radomly colored from a set of (1 + o(1)) colors, with p = (1+o(1))log, we show that oe ca almost always fid a Hamilto cycle which has the additioal property that all edges are distictly colored (raibow). 1 Itroductio Hamilto cycles occupy a positio of cetral importace i graph theory, ad are the subject of coutless results. I the cotext of radom structures, much research has bee doe o may aspects of Hamiltoicity, i a variety of radom structures. See, e.g., ay of [3, 4, 5, 19, 24] cocerig Erdős-Réyi radom graphs ad radom regular graphs, ay of [6, 14, 20, 21] regardig directed graphs, or ay of the recet developmets [9, 11, 12, 13] o uiform hypergraphs. I this paper we cosider the existece of raibow Hamilto cycles i edge-colored graphs. (A set S of edges is called raibow if each edge of S has a differet color.) There are two geeral types of results i this area: existece whp 1 uder radom colorig ad guarateed existece uder adversarial colorig. Whe cosiderig adversarial(worst-case) colorig, the guarateed existece of a raibow structure is called a Ati-Ramsey property. Erdős, Nešetřil, ad Rödl [10], Hah ad Thomasse [17] ad Albert, Frieze, ad Reed [1] (correctio i Rue [25]) cosidered colorigs of the edges of the complete graph K where o color is used more tha k times. It was show i [1] that if k /64, the there must be a raibow Hamilto cycle. Cooper ad Frieze [7] proved a radom graph threshold for this property to hold i almost every graph i the space studied. There is also a history of work o radom colorig (see, e.g., ay of [7, 8, 15, 16]), ad it has recetly become apparet that this radom settig may be of substatial utility. Ideed, a result of Jaso ad Wormald [16] o raibow Hamilto cycles i radomly edge-colored radom regular Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA 15213, ala@radom.math.cmu.edu. Research supported i part by NSF award DMS Departmet of Mathematical Scieces, Caregie Mello Uiversity, Pittsburgh, PA 15213, ploh@cmu.edu. 1 A sequece of evets E is said to occur with high probability (whp) if lim PrE = 1. 1

2 graphs played a cetral role i the recet determiatio of the threshold for loose Hamiltoicity i radom 3-uiform hypergraphs by Frieze [11]. Roughly speakig, a hyperedge (triple of vertices) ca be ecoded by a ordiary edge (pair of vertices), together with a color. Hece, a radom 3-uiform hypergraph gives rise to a radomly edge-colored radom graph. We will discuss this further i Sectio 4. Let us ow focus o the radom colorig situatio, where we cosider the followig model. Let G,p,κ deote a radomly colored radom graph, costructed o the vertex set [] by takig each edge idepedetly with probability p, ad the idepedetly colorig it with a radom color from the set [κ]. We are iterested i coditios o,p,κ which imply that G,p,κ cotais a raibow Hamilto cycle whp. The startig poit for our preset work is the followig theorem of Cooper ad Frieze. Theorem. (See [8], Theorem 1.1.) There exist costats K 1 ad K 2 such that if p > K 1log κ > K 2, the G,p,κ cotais a raibow Hamilto cycle whp. ad The aim of this paper is to substatially stregthe the above result by provig the followig theorem: Theorem 1.1. If p = (1+ǫ)log 125 ad κ = (1+θ), where ǫ,θ > loglog, the G,p,κ cotais a raibow Hamilto cycle whp. To discuss the tightess of our mai theorem, let us recall the threshold for Hamiltoicity i G,p, established by Komlós ad Szemerédi [19]. We fid that we must have p > log+loglog+ω() with ω(), or else the uderlyig ucolored G,p will ot eve be Hamiltoia. We also eed at least colors to appear o the edges i order to have eough colors for a raibow Hamilto cycle. Note that the earlier result came withi a costat factor of both of these miimum requiremets. Our theorem drives both costats dow to be best possible up to first order. We allow our error terms ǫ ad θ to decrease slowly, although we do ot expect our costraits o them to be optimal. Our discussio above shows that the trivial lower boud for ǫ is aroud loglog log. The, if p log, we eed at least +Ω(1/2 ) colors just to esure that whp at least distict colors occur o the m 1 1 2log edges i the graph; hece, the trivial lower boud for θ is aroud. We leave further exploratio to future work, ad highlight a potetial aswer i our coclusio. This paper is orgaized as follows. Sectio 2 provides a outlie of our proof. The proofs of the mai steps follow i the sectio thereafter. We coclude i Sectio 5 with some remarks ad ope problems. The followig (stadard) asymptotic otatio will be utilized extesively. For two fuctios f() ad g(), we write f() = o(g()), g() = ω(f()), or f() g() if lim f()/g() = 0, ad f() = O(g()) or g() = Ω(f()) if there exists a costat M such that f() M g() for all sufficietly large. We also write f() = Θ(g()) if both f() = O(g()) ad f() = Ω(g()) are satisfied. All logarithms will be i base e Proof of Theorem 1.1: high level descriptio I our proofs below we will assume that ǫ,θ are sufficietly small so that various iequalities are true. I additio we will assume that ǫ,θ > 125 loglog. We will implicitly assume throughout (whe coveiet) that they are sufficietly small. Our proof proceeds i three phases, so our parameters come i threes. Let us arbitrarily partitio the 2

3 κ = (1+θ) colors ito three disjoit groups C 1 C 2 C 3, with sizes C 1 = θ 1, C 2 = (1+θ 2 ), C 3 = θ 3. We will aalyze the radom edge geeratio i three stages, so we defie the probabilities p 1 = ǫ 1log 2, The ǫ s ad θ s are defied by the relatios ǫ 1 = ǫ 2 = ǫ 3 = ǫ 3, p 2 = (1+ǫ 2)log, p 3 = ǫ 3log 2 2. θ 1 = θ 3 = mi { θ 3, ǫ } 2, θ 2 = θ θ 1 θ 3. (1) 4 (We would have take θ 1 = θ 2 = θ 3 = θ 3, except that Lemma 3.9 requires θ 1 +θ 3 ǫ 2 2.) 2.1 Uderlyig digraph model It is more coveiet for our etire argumet to work with directed graphs, as this will allow us to coserve idepedece. Recall that D,p is the model where each of the ( 1) possible directed edges appears idepedetly with probability p. We geerate a radom colored udirected graph via the followig procedure. First, we idepedetly geerate three digraphs D1 = D,p 1, D2 = D,p 2, ad D3 = D,p 3, ad color all of the directed edges idepedetly ad uiformly at radom from the full set of colors. We ext use the Di to costruct a colored udirected graph G, by takig the udirected edge uv if ad oly if at least oe of uv or vu appear amog the Di. The colors of the udirected edges are iherited from the colors of the directed edges, i the priority order D1, D 2, D 3. Specifically, if uv or vu appear already i D1, the uv takes the color used i D 1 eve if uv or vu appear agai i D3, say. I the evet that both uv ad vu appear i D1, the color of uv is used for uv with probability 1/2, ad the color of vu is used otherwise. Similarly, if either of uv or vu appear i D1, but uv appears i both D2 ad D 3, the color used i D 2 takes precedece. It is clear that the resultig colored graph G has the same distributio as G,p,κ, with 2.2 Partitioig by color p = 1 (1 p 1 ) 2 (1 p 2 ) 2 (1 p 3 ) 2 = (1+ǫ+O(ǫ 2 )) log. I each of our three phases, we will use oe group of edges ad oe group of colors. Sice each D i cotais edges colored from the etire set C 1 C 2 C 3, for each i we defie D i D i to be the spaig subgraph cosistig of all directed edges whose color is i C i. Our fial udirected graph is geerated by superimposig directed graphs ad disregardig the directios. Cosequetly, we do ot eed to hoor the directios whe buildig Hamilto cycles. To accout for this, we defie three correspodig colored udirected graphs G 1, G 2, ad G 3. These will be edge-disjoit, respectig priority. The first, G 1, is costructed as follows. For each pair of vertices u,v with uv D 1 but vu D 1, place uv i G 1 i the same color as uv. If both uv ad vu are i D 1, we still place the edge uv i G 1, but radomly select either the color of uv or of vu. However, if uv D 1 but vu D 1 \ D 1, 3

4 the uv is oly placed i G 1 with probability 1/2; if it is placed, it iherits the color of uv. Note that this costructio precisely captures all udirected edges arisig from D1, usig colors i C 1. We are less careful with G 2, as our argumet ca afford to discard all edges that arise from multiply covered pairs. Specifically, we place uv G 2 if ad oly if uv D 2 \D1 ad vu D1 D 2. As the pair {u,v} is ow spaed by oly oe directed edge i D2, the udirected edge uv iherits that uique color. We defie G 3 similarly, placig uv G 3 if ad oly if uv D 3 \(D1 D 2 ) ad vu D1 D 2 D 3. I this way, we create three edge-disjoit graphs G i. By our observatios i the previous sectio, we may ow focus o fidig a (udirected) raibow Hamilto cycle i G 1 G 2 G 3. Importatly, ote that i terms of geeratig colored udirected edges, the digraph D1 has higher priority tha D2 or D 3. So, for example, the geeratio of G 1 is ot affected by the presece or absece of edges from D2 or D Mai steps We geerally prefer to work with G i ad D i istead of Di because we are guarateed that the edge colors lie i the correspodig C i. This allows us to build raibow segmets i separate stages, without worryig that we use the same color twice. Let d + i (v) deote the out-degree of v i D i. We ow defie a set S of vertices that eed special treatmet. We first let S 0 = S 0,1 S 0,2 S 0,3, where { S 0,1 = v : d + 1 (v) ǫ } 1θ 1 20 log (2) S 0,2 = {v : d +2 (v) 120 } log (3) { S 0,3 = v : d + 3 (v) ǫ } 3θ 3 20 log. (4) Also, defie γ = mi { 1 4, 1 4 ǫ 1θ 1, 1 4 ǫ } 3θ 3, ad ote that the costraits o ǫ,θ i Theorem 1.1 imply the boud ( γ > ) > loglog Lemma 2.1. With probability 1 o( 1 ), the set S 0 satisfies S γ. 1 loglog. (5) The vertices i S 0 are delicate because they have low degree. We also eed to deal with vertices havig several eighbors i S 0. For this, we defie a sequece of sets S 0,S 1,...,S t i the followig way. Havig chose S t, if there is still a vertex v S t with at least 4 out-eighbors i S t (i ay of the graphs D 1, D 2, or D 3 ), we let S t+1 = S t {v} ad cotiue. Otherwise we stop at some value t = T ad take S = S T. Lemma 2.2. With probability 1 o( 1 ), the set S cotais at most 1 γ vertices. To take care of the dagerous vertices i S, we fid a collectio of vertex disjoit paths Q 1,Q 2,...,Q s, s S such that (i) each path uses udirected edges i G 2, (ii) all colors which 4

5 appear o these edges are distict, (iii) all iterior vertices of the paths are vertices of S, (iv) every vertex of S appears i this way, ad (v) the edpoits of the paths are ot i S. Let us say that these paths cover S. Lemma 2.3. The graph G 2 cotais a collectio Q 1,Q 2,...,Q s of paths that cover S whp. The ext step of our proof uses a radom greedy algorithm to fid a raibow path of legth close to, avoidig all of the previously costructed Q i. Lemma 2.4. The graph G 2 cotais a raibow path P of legth = 3 log whp. Furthermore, P is etirely disjoit from all of the Q i, ad all colors used i P ad the Q i are distict ad from C 2. Let U be the vertices outside P. (Note that U cotais all of the paths Q i.) I order to lik the vertices of U ito P, we split P ito short segmets, ad use the edges of G 3 to splice U ito the system of segmets. We will later use the edges of G 1 to lik the segmets back together ito a raibow Hamilto cycle, so care must be take to coserve idepedece. The followig lemma merges all vertices of U ito the collectio of segmets, ad prepares us for the fial stage of the proof. Here, d + 1 (v;a) deotes the umber of D 1-edges from v to a set A. Let { L = max 10 e 40 ǫ 3 θ 3, 7 }, (6) θ 1 ad ote that our coditios o ǫ,θ i Theorem 1.1, together with (1), imply that ǫ 3 θ 3 > ( ) loglog > 434, so we have loglog { L < max 10 e loglog,7 12 } loglog < 10 log. (7) 100 Lemma 2.5. With probability 1 o(1), the etire vertex set ca be partitioed ito segmets I 1,...,I r, with r = (1 o(1)) L, such that the edges which appear i the segmets all use differet colors from C 2 C 3. The segmet edpoits are further partitioed ito A B, with each segmet havig oe edpoit i A ad oe i B, such that every a A has d + 1 (a;b) ǫ 1θ 1 200L log, ad every b B has d + 1 (b;a) ǫ 1θ 1 200L log. All of the umeric values d+ 1 (a;b) ad d+ 1 (b;a) have already bee revealed, but the locatios of the correspodig edges are still idepedet ad uiform over B ad A, respectively. The fial step liks together the segmets I 1,...,I r usig distictly-colored edges from G 1. For this, we create a auxiliary colored directed graph Γ, which has oe vertex w k for each segmet I k. There is a directed edge w j w k Γ if there is a edge e G 1 betwee the B-edpoit of I j ad the A-edpoit of I k ; it iherits the color of e. Sice all colors of edges i Γ are from C 1, it therefore suffices to fid a raibow Hamilto directed cycle i Γ. We will fid this by coectig Γ with a well-studied radom directed graph model. Defiitio 2.6. The d-i, d-out radom directed graph model D d-i,d-out is defied as follows. Each vertex idepedetly chooses d out-eighbors ad d i-eighbors uiformly at radom, ad all resultig directed edges are placed i the graph. Due to idepedece, it is possible that a vertex u selects v as a out-eighbor, ad v also selects u as a i-eighbor. I that case, istead of placig two repeated edges uv, place oly oe. 12 5

6 Istead of provig Hamiltoicity from scratch, we apply the followig theorem of Cooper ad Frieze. Theorem 2.7. (See [6], Theorem 1.) The radom graph D 2-i,2-out cotais a directed Hamilto cycle whp. This result does ot take colors ito accout, however. Fortuately, i equatio (6), we defie L to be large eough to allow us to select a subset of G 1 -edges which is itself already raibow. The aalysis of this procedure is the heart of the proof of the fial step. Lemma 2.8. The colored directed graph Γ cotais a raibow directed Hamilto cycle whp. Sice each directed edge of Γ correspods to a udirected G 1 -edge from a B-edpoit of a segmet to a A-edpoit of aother segmet, a directed Hamilto cycle i Γ correspods to a Hamilto cycle likig all of the segmets together. Lemma 2.8 establishes that it is possible to choose theselikigedgesas araibow set fromc 1. The edges withi the segmets werethemselves colored from C 2 C 3, so the result is ideed a raibow Hamilto cycle i the origial graph, as desired. 3 Proofs of itermediate lemmas I the remaider of this paper, we prove the lemmas stated i the previous sectio. Although the first lemma is fairly stadard, we provide all details, ad use the opportuity to formally state several other well-kow results which we apply agai later. 3.1 Proof of Lemma 2.1 Our first lemma cotrols the umber of vertices whose degrees i D i are too small. Recall from Sectio 2.1 that the Di are idepedetly geerated. Their edges are the idepedetly colored, ad the edges of Di which receive colors from C i are collected ito D i. (Priorities oly take effect whe we form the G i i Sectio 2.2.) Therefore, the out-degrees d + i (v) of vertices v i D i are distributed as ) ( d + 1 ( 1,p (v) Bi θ 1 1 Bi 0.99, 0.49ǫ ) 1θ 1 log 1+θ 1 +θ 2 +θ 3 ) ( d + 2 ( 1,p (v) Bi 1+θ 2 2 Bi 0.99, 0.49log ) 1+θ 1 +θ 2 +θ 3 ) d + 3 ( 1,p (v) Bi θ 3 3 Bi 1+θ 1 +θ 2 +θ 3 ( 0.99, 0.49ǫ 3θ 3 log ). Here ad i the remaider, we will say that π 1 π 2 for distributios π 1,π 2 if we ca fid a couplig of radom variables (X,Y) such that X Y ad X π 1,Y π 2. Thus the expected size of S 0 satisfies E[ S 0 ] (ρ 1 +ρ 2 +ρ 3 ), 6

7 where [ ( ρ 1 = P Bi 0.99, 0.49ǫ ) 1θ 1 log ǫ ] 1θ 1 log 20 [ ( ρ 2 = P Bi 0.99, 0.49log ) log ] 20 [ ( ρ 3 = P Bi 0.99, 0.49ǫ ) 3θ 3 log ǫ ] 3θ 3 log. 20 We will repeatedly use the followig case of the Cheroff lower tail boud, which we prove with a appropriate explicit costat. Lemma 3.1. The followig holds for all sufficietly large mq, where m is a positive iteger ad 0 < q < 1 is a real umber. P [Bi(m,q) 19 ] mq < e 0.533mq. Proof. Calculatio yields P [Bi(m,q) 19 ] mq = mq/9 k=0 ( ) m q k (1 q) m k < k mq/9 k=0 ( emq ) ke 8 9 mq k where ( emq) k ( k = 1bycovetiofork = 0. Thefuctio C ) k k = exp{k(logc logk)}isicreasig i k i the rage 0 < k < C/e. Thus P [Bi(m,q) 19 ] mq < 2mq ( ) emq mq/9 e 8 9 mq 9 mq/9 = 2mq (9e) mq/9 e 8 9 mq 9 = e mq(1 9 log9e 8 9 +o(1)) < e 0.533mq, as claimed. Returig to the proof of Lemma 2.1, we observe that sice 1 20 < , a direct applicatio of Lemma 3.1 ow gives [ ( ρ 2 < P Bi 0.99, 0.49log ) 19 ] log < e log < Asimilarargumetestablishesthatρ 1 < 0.258ǫ 1θ 1 adρ 3 < 0.258ǫ 3θ 3. ThisprovesthatE[ S 0 ] = o( 1 γ ), where we recall our defiitio γ = mi { 1 4, 1 4 ǫ 2θ 2, 1 4 ǫ } 3θ 3. We complete the proof of the lemma by showig that S 0 is cocetrated aroud its mea. For this, we use the Hoeffdig-Azuma martigale tail iequality applied to the vertex exposure martigale (see, e.g., [2]). Recall that a martigale is a sequece X 0,X 1,... of radom variables such that each coditioal expectatio E[X t+1 X 0,...,X t ] is precisely X t. 7

8 Theorem 3.2. Let X 0,...,X be a martigale, with bouded differeces X i+1 X i C. The for ay λ 0, } P[X X 0 +λ] exp { λ2 2C 2. Here we cosider S 0 to be a fuctio of Y 1,Y 2,...,Y where Y k deotes the set of edges jk, kj D1 D 2 D 3, j < k. The sequece X t = E[ S 0 Y 1,...,Y t ] is called the vertex-exposure martigale. There is a slight problem i that the worst-case Lipschitz value for chagig a sigle Y k ca be too large, while the average case is good. There are various ways of dealig with this. We will make a small chage i D = D1 D 2 D 3. Let ˆD be obtaied from D by reducig every degree below 5log. We do this i vertex order v = 1,2,..., ad delete edges icidet with v i descedig umerical order. We ca show that this usually has o effect o D. Lemma 3.3. With probability 1 o( 1 ), every vertex i G,p with p < 1.1log 5log. has degree at most Proof. The probability that a sigle vertex has degree at least 5log is ( )( 1.1log P[Bi( 1,p) 5log] 5log ( e 5log 1.1log ( ) 1.1e 5log = 5 = 2.57, ) 5log ) 5log so a uio boud over all vertices gives the result. [ˆD ] Therefore, P = D = 1 o( 1 ), ad so if we let Ẑ = Ŝ0 be the size of the correspodig set evaluated i ˆD ], we obtai E[Ẑ = E[ S 0 ]+o(1) = o( 1 γ ). Furthermore, chagig a Y k ca oly chage Ẑ by at most 15log. So, we have [ P Ẑ i E [Ẑi ]+ 1 ] { } 4 1 γ exp 2 2γ /16 2(15log) 2 < o( 1 ), completig the proof of Lemma Proof of Lemma 2.2 We use the followig stadard estimate to cotrol the desities of small sets. Lemma 3.4. With probability 1 o( 1 ), i D,p with p < log, every set S of fewer tha 4 e 4 log 2 vertices satisfies e(s) < 2 S. Here, e(s) is the umber of directed edges spaed by S. Proof. Fix a positive iteger s < 4, ad cosider sets of size s. We may assume that s 2, e 4 log 2 because a sigle vertex caot iduce ay edges. The expected umber of sets S with S = s ad 8

9 e(s) 2s is at most ( ) s ( )( ) s 2 log 2s ( e ) ) s es 2 2s ( log ( 2s s 2s ( = s e3 log 2 ) s. 4 ) 2s It remais to show that whe this boud is summed over all 2 s < 4, the result is still e 4 log 2 o( 1 ). Ideed, for each 2 s 2log, the boud is at most O ( log 6 ), so the total cotributio 2 from that part is oly O ( log 7 ) = o( 1 ). O the other had, for each 2log < s < 4 2 e 4 log 2, the boud is at most ( 4 e 4 log 2 e3 log 2 ) 2log = 4 ( ) 1 2log = 2. e Thus the total cotributio from 2log < s < 4 e 4 log 2 is at most o( 1 ), as desired. We are ow ready to boud the size of the set S which was created by repeatedly absorbig vertices with may eighbors i S 0. Proof of Lemma 2.2. We actually prove a stroger statemet, which we will eed for Lemma 3.7. Suppose we have a iitial S 0 satisfyig S 0 < γ, as esured by Lemma 2.1. Cosider a sequece S 0,S 1,S 2,... where S t+1 is obtaied from S t by addig a vertex v / S t for which d + i (v;s t) 3 for some i. Note that whe this process stops, the fial set S will cotai the set S which by our defiitio is obtaied by addig vertices with degree at least 4 ito previous S t. So, suppose that this process cotiues for so log that some S t reaches 1 γ = o ( ) log 2. Note that t γ. Sice each step itroduces at least 3 edges, we must have e(s t) 3t 2 1 γ = 2 S t. Lemma 3.4 implies that ca oly happe with probability o( 1 ). (By costructio, D 1 D 2 D 3 is a istace of D,q for some q < log ). 3.3 Proof of Lemma 2.3 I this sectio, we show that for each vertex v S, we ca fid a disjoit G 2 -path Q cotaiig v which starts ad eds outside S. We also eed all colors appearig o these edges to be differet. Sice we are workig i a regime where degrees ca be very small, we eed to accommodate the most delicate vertices first. Specifically, let S 0,0 be the set of all vertices with d 2 (v) 1 10 log, where d 2 (v) is the degree of v i G 2. Although S 0,0 will typically ot be etirely cotaied withi S 0,2, we ca show that it is still usually quite small. Lemma 3.5. We have S 0,0 < 0.48 whp. Proof. By costructio, G 2 G,q2, where q 2 = 2p 2 (1 p 2 ) 1+θ 2 1+θ 1 +θ 2 +θ 3 (1 p 1 ) 2, (8) because the first factor is the probability that exactly oe of uv or vu appears i D2, the secod factor is the probability that it receives a color from C 2, ad the third factor is the probability that 9

10 either uv or vu appear i D1. Hece for a fixed vertex v, its relevat degree i G 2 is distributed as ( d 2 (v) Bi( 1,q 2 ) Bi 0.99, 0.99log ). Sice 1 10 < , Lemma 3.1 implies that P [d 2 (v) 110 ] [ ( log < P Bi 0.99, 0.99log ) 19 ] log < e log < (9) Therefore, E[ S 0,0 ] < 0.522, ad Markov s iequality yields the desired result. We have show that vertices of S 0,0 are few i umber. Our ext result shows that they are also scattered far apart. This will help us whe we costruct the coverig paths, by prevetig paths from collidig. Lemma 3.6. Let dist 2 (v,w) deote the distace betwee v ad w i G 2. The, whp, every pair v,w S 0,0 satisfies dist 2 (v,w) 5. Proof. Recall that G 2 G,q2 with q 2 defied as i (8). Cosider a fixed pair of vertices v,w. For a fixed sequece of k 4 itermediate vertices x 1,x 2,...,x k, let us boud the probability q that v,w both have d log, ad all the edges vx 1,x 1 x 2,x 2 x 3,...,x k w appear i G 2. First expose the edges vx 1, x 1 x 2,..., x k w, ad the expose the edges betwee v ad []\{v,x 1,...,x k,w}, ad betwee w ad that set. This gives the followig boud o our probability q: ( ) 1.01log k+1 ( q P[ Bi 2 k, 0.99log ) log ] 2 (10) 10 A calculatio aalogous to (9) bouds the Biomial probability by 0.52, so takig a uio boud over all O( k ) choices for the x i, for all 0 k 4, we fid that for fixed v,w, the probability that dist 2 (v,w) < 5 is at most 4 ( ) 1.01log k+1 O( k ) ( 0.52) 2 < 2.04+o(1). k=0 Therefore, a fial uio boud over the O( 2 ) choices for v,w completes the proof. The previous result will help us cover vertices i S 0,0 with G 2 -paths. However, the objective of this sectio is to cover all vertices of S. Although the aalogue of Lemma 3.6 does ot hold for S, it is still possible to prove that S is sparsely coected to the rest of the graph. Recall from (5) that γ = mi { 1 4, 1 4 ǫ 1θ 1, 1 4 ǫ } 3θ 3 > 1 loglog. Lemma 3.7. With respect to edges of G 2, every vertex v is adjacet to at most 2 γ whp. (This applies whether or ot v itself is i S.) vertices i S 10

11 Proof. Fix a vertex v. Let S be the set obtaied by costructig the aalogous sequeces to S 0,1, S 0,2, S 0,3, S 1,... o the graph iduced by []\{v}, where S 0,i, i = 1,2,3 is defied as i (2) (4), but the S t+1 are obtaied by addig vertices with at least 3 (ot 4) D i-out-eighbors i S t. Clearly, S cotais S\{v}, because the effect of igorig v is compesated for by usig 3 istead of 4. The advatage of usig S istead of S is that S ca be geerated without exposig ay edges icidet to v. As we will take a fial uio boud over the choices of v, it therefore suffices to show that with probability 1 o( 1 ), the particular vertex v has at most 2 γ eighbors i S. For this, we expose all edges of D1 D 2 D 3 that are spaed by []\{v}. Recall that our proof of Lemma 2.2 already absorbed vertices with 3 out-eighbors (istead of 4), so we have S 1 γ with probability 1 o( 1 ). It remais to cotrol the umber of edges betwee v ad S, so we ow expose all edges of D1 D 2 D 3. The G 2-edges there appear idepedetly with probability q 2 as defied i (8), so the probability that at least 2 γ edges appear is at most [ P Bi ( ] ( )( 1 γ ) 2 1 γ 1.01log,q 2 γ 2/γ ( e 1 γ 1.01log 2/γ ( ) 2γlog 2/γ < γ ) 2/γ ) 2/γ < (2log)2loglog 2 = o( 1 ). (11) Takig a fial uio boud over all iitial choices for v completes the proof. We will cover each vertex v S with a G 2 -path by joiig two G 2 -paths of legth up to 2, each origiatig from v. It is therefore coveiet to exted the previous result by oe further iteratio. Corollary 3.8. With respect to edges of G 2, every vertex v is withi distace two of at most ( 2 γ vertices i S whp. (This applies whether or ot v itself is i S.) Proof. Fix a vertex v. Costruct S i the same way as i the proof of Lemma 3.7, exposig oly edges spaed by []\{v}. Usig oly those exposed edges, let T []\{v} be the set of all vertices i S or adjacet to S via edges from G 2. By Lemma 3.3, the maximum degree of G 2 \{v} is at most 5log with probability 1 o( 1 ), so T 1 γ 5log. A similar calculatio to (11) the shows that with probability 1 o( 1 ), v has at most 2 γ eighbors i T. Takig a uio boud over all v, ad combiig this with Lemma 3.7, we coclude that whp, every vertex has at most 2 γ eighbors i S N(S), ad each of them has at most 2 γ eighbors i S. This implies the result. We are ow ready to start coverig the vertices of S with disjoit raibow G 2 -paths. The most delicate vertices are those i S 0,0, because by defiitio all other vertices already have G 2 -degree at least 1 10 log. Naturally, we take care of S 0,0 first. Lemma 3.9. The colored graph G 2 cotais a raibow collectio Q 1,Q 2,...,Q s of disjoit paths that cover S S 0,0 whp. ) 2 11

12 Proof. We coditio o the high-probability evets i Lemmas 3.5, 3.6, ad 3.7, ad use a greedy algorithm to cover each v S S 0,0 with a path of legth 2, 3, or 4. Recall that G 2 G,q2, where q 2 was specified i (8). We ca boud q 2 by q 2 > (1+ǫ 2 ) log ( 1 log ) ( (1 θ 1 θ 3 ). 1 log ) 2. Sice equatio (1) esures that θ 1 + θ 3 ǫ 2 2, ad our coditios o Theorem 1.1 force ǫ 2 > loglog loglog log, the miimum degree i G 2 is at least two whp, see e.g. [3]. Coditio o this as well. Now cosider a vertex v S S 0,0, ad let x 1 ad x 2 be two of its eighbors. If both x i are already outside S, the we use x 1 vx 2 to cover v. Otherwise, suppose that x 1 is still i S. Sice we coditioed o vertices i S 0,0 beig separated by distaces of at least 5 (Lemma 3.6), x 1 caot be i S 0,0, so it has at least 1 10 log G 2-eighbors. These caot all be i S, because we coditioed o the fact that every vertex has fewer tha 2 γ < 2loglog eighbors i S (Lemma 3.7). So, we ca pick oe, say y 1, such that y 1 x 1 v is a path from outside S to v. A similar argumet allows us to cotiue the path from v to a vertex outside S i at most two steps. Therefore, there is a collectio of paths of legth 2 4 coverig each vertex i S S 0,0. They are all disjoit, sice we coditioed o vertices of S 0,0 beig separated by distaces of at least 5. At this poit, we have exposed all G 2 -edges spaed by S ad its eighbors, but the oly thig we have revealed about their colors is that they are all i C 2. Now expose the precise colors o all edges of these paths. Sice we coditioed o S 0,0 < 0.48 (Lemma 3.5), the total umber of edges ivolved is at most < The umber of colors i C 2 is (1+θ 2 ), so by a simple uio boud the probability that some pair of edges receives the same color i C 2 is at most ( ) (1+θ 2 ) = o(1). Therefore, the coverig paths form a raibow set whp, as desired. We have ow covered the most dagerous vertices of S. The remaider of this sectio provides a argumet which covers all other vertices i S. Proof of Lemma 2.3. Coditio o the high-probability evets of Lemmas 2.2, 3.7, 3.9, ad Corollary 3.8. We have already covered all vertices i S S 0,0 with disjoit raibow paths of legths up to four (Lemma 3.9). We cover the rest of the vertices i S\S 0,0 with paths of legth two, usig a simple iterative greedy algorithm. Ideed, suppose that we are to cover a give vertex v S\S 0,0. Sice it is ot i S 0,0, it has G 2 -degree at least 1 10 log, ad at most 2 γ of these eighbors ca be withi S (Lemma 3.7). Furthermore, we ca show that at most 2( 2 γ )2 of v s eighbors outside S ca already have bee used by coverig paths. Ideed, for each eighbor w S of v that was used by a previous coverig path, we could idetify a vertex x S adjacet to w which was part of that coverig path. Importatly, x is withi distace two of v, so the collectio of all x obtaiable i this way is of size at most ( 2 γ )2, as we coditioed o Corollary 3.8. Sice every coverig path uses exactly two vertices outside S, the total umber of such w is at most 2( 2 γ )2. Puttig everythig together, we coclude that the umber of usable G 2 -edges emaatig from v is at least 1 10 log 2 ( ) 2 2 γ 2 > 1 γ 11 log. 12

13 Expose the colors (ecessarily from C 2 ) which appear o these G 2 -edges. Of the total of (1+θ 2 ) available, we oly eed to avoid at most 4 S which have already bee used o previous coverig paths. Sice we coditioed o S 1 γ (Lemma 2.2), this is at most 4 1 γ colors to avoid. We oly eed to have two ew colors to appear amog this collectio i order to add a ew raibow path of legth two coverig v. Takig aother uio boud, we fid that the probability that at most oe ew color appears is at most (1+θ 2 ) ( 4 1 γ ) log = o( 1 ). (1+θ 2 ) Here, the first factor of (1+θ 2 ) correspods to the umber of ways to choose the ew color to add (or oe at all). Sice we oly ru our algorithm for o() iteratios (oce per vertex i S \S 0,0 ), we coclude that whp we ca cover all vertices of S with disjoit raibow G 2 -paths. 3.4 Proof of Lemma 2.4 I this sectio, we costruct a raibow G 2 -path which cotais most of the vertices of the graph, but avoids all coverig paths from the previous sectio. I order to carefully track the idepedece ad exposure of edges, recall from Sectio 2.2 that G 2 is determiistically costructed from the radom directed graphs D1, D 2, ad D 3. Let us cosider the geeratio of the D i to be as follows. The probability that the directed edge θ vw appears i D 1 is p θ 1 +θ 2 +θ 3, so we expose each ) D 1 -out-degree d + 1 ( 1,p (v) by idepedetly samplig from the Bi θ θ 1 +θ 2 +θ 3 distributio. Importatly, we do ot reveal the locatios of the out-eighbors. Similarly, for D 2 ad D 3, we expose all out-degrees d + 2 (v) ad d+ 3 (v), each sampled from the appropriate Biomial distributio. By Lemma 3.3, all d + i (v) 5log whp; we coditio o this. Note that from this iformatio, we ca later fully geerate (say) D 1 ad D1 as follows. At each vertex v, we idepedetly choose d + 1 (v) out-eighbors uiformly at radom. This will determie all D 1 -edges. Next, for every edge which is ot part of D 1, idepedetly sample it to be part of D1 \D ( 1 with probability p 1 1 θ 1 ) 1+θ 1 +θ 2 +θ 3. This will determie all D 1 edges, ad a similar system will determie all edges of D2 ad D 3. Returig to the situatio where oly the d + i (v) have bee exposed, we the costruct the S 0,i by collectig all vertices whose d + i (v) are too small, ad build the sequece S 0,S 1,S 2,...,S t. I each iteratio of that process, we go over all vertices which are ot yet i the curret S t. At each v, we expose all D i -edges icidet to S t. For this sectio, we will oly care about the D 2 -out-edges from v S (iitially couted by d + 2 (v)) that are ot cosumed i this process. Fortuately, at each exposure stage, there is a clear distributio o the umber of these out-edges that are cosumed toward S t, ad this will oly affect the umber, ot the locatio, of the out-edges which are ot cosumed. Therefore, after this procedure termiates, we will have a fial set S, ad the set of revealed (directed) edges is precisely those edges spaed by S, together with all those betwee S ad V 1 = [] \ S. This set of revealed edges is exactly what is required to costruct the coverig paths Q 1,...,Q s i Lemma 2.3. Withi V 1, the precise locatios of the edges are ot yet revealed. Istead, for each vertex v V 1, there is ow a umber d 2 (v), correspodig to the umber of D 2 -out-edges from v to vertices outside S. 13

14 We ow make two crucial observatios. First, the distributios of where these edpoits lie are still idepedet ad uiform over V 1. Secod, every d 2 (v) 5log ad d 2 (v) 1 20 log 3 > 1 21 log, because if there were 4 out-edges from v to S, the v should have bee absorbed ito S durig the process. This abudace of idepedece makes it easy to aalyze a simple method for fidig a log path, based o a greedy algorithm with backtrackig. (This procedure is similar to that used i [23] by Feradez de la Vega.) Ideed, the most straightforward attempt would be to start buildig a path, ad at each iteratio expose the out-edges of the fial edpoit, as well as their colors. If there is a optio which keeps the path raibow, we would follow that edge, ad repeat. If ot, the we should backtrack to the latest vertex i the path which still has a optio for extesio. We formalize this i the followig algorithm. Particularly dagerous vertices will be coded by the color red (ot related to the colors of the edges i the G,p,κ ). Let V 2 V 1 be the set of all vertices which are ot ivolved i the coverig paths Q i. We will fid a log G 2 -path withi V 2 which avoids all of the coverig paths. Algorithm. 1. Iitially, let all vertices of V 2 be ucolored, ad select a arbitrary vertex v V 2 to use as the iitial path P 0 = {v}. Let U 0 = V 2 \ {v}. This is the set of utouched vertices. Let R 0 =. This will cout the red vertices. 2. Now suppose we are at time t. If U t < 2 3, termiate the algorithm. log 3. If the fial edpoit v of P t is ot red, the expose the first 1 2 d 2 (v) of v s D 2-out-eighbors. If oe of them lies i U t, via a edge color ot yet used by P t or ay of the coverig paths Q i, the color v red, settig U t+1 = U t, P t+1 = P t, ad R t+1 = R t {v}. Otherwise, arbitrarily choose oe of the suitable out-eighbors w U t. Set U t+1 = U t \{w}. Expose whether vw D1, wv D1, or wv D2. If oe of those three directed edges are preset, the add w to the path, settig P t+1 = P t {w} ad R t+1 = R t. Otherwise, color both v ad w red, ad set P t+1 = P t ad R t+1 = R t {v,w}. 4. If the fial edpoit v of P t is red, the expose the secod 1 2 d 2 (v) of v s D 2-out-eighbors. First suppose that oe of them lies i U t, via a edge color ot yet used by P t or ay of the coverig paths Q i. I this case, fid the last vertex v of P t which is ot red, color it red, ad make it the ew termius of the path. That is, set U t+1 = U t, let P t+1 be P t up to v, ad set R t+1 = R t {v }. If v did ot exist (i.e., all vertices of P t were already red), the istead let v be a arbitrary vertex of U t ad restart the path, settig P t+1 = {v }, R t+1 = R t, U t+1 = U t \{v }. O the other had, if v has a suitable out-eighbor w U t, the set U t+1 = U t \{w}. Expose whether vw D1, wv D1, or wv D2. If oe of those three directed edges are preset, the add w to the path, settig P t+1 = P t {w} ad R t+1 = R t. Otherwise, color w red, fid the last vertex v of P t which is ot red, ad follow the remaider of the first paragraph of this step. The key observatio is that the fial path P T cotais every o-red vertex which lies i V 2 \U T. 14

15 Sice Lemmas 2.2 ad 2.3 imply that V 2 3 S 3 1 γ, ad we ru util U T < 2 3 log, Lemma 2.4 therefore follows from the followig boud ad the fact that γ > 1/loglog 3 log. Lemma The fial umber of red vertices is at most e log whp. Proof. The color red is applied i oly two situatios. The first is whe we expose whether ay of vw D1, wv D1, or wv D2 hold. To expose whether vw D1, we reveal whether vw D 1, usig the previously exposed value of d 1 (v), which we already coditioed o beig at most 5log. Sice v s D 1 -out-eighbors are uiform, the probability that vw D 1 is at most. If it 5log (1 o(1)) is ot i D 1, the probability that it is i D1 \ D 1 is bouded by log by the descriptio at the begiig of Sectio 3.4. The aalysis for the other two cases are similar, so a uio boud gives that the chace that ay of vw D1, wv D1, or wv D2 7log hold is at most 3. Note that this occurs at most times, because each istace reduces the size of U t by 1. Hece the expected umber of red vertices of this type is at most O(log), which is of much smaller order tha e Θ( 3 log). The other situatio i which red is applied comes immediately after the failed exposure of some k = 1 2 d 2 (v) > 1 42 log D 2-out-eighbors, i either of Steps 3 or 4. Failure meas that all k of them either fell outside U t, or had edge colors already used i P t or some coverig path Q i. Step 2 cotrols U t 2 3 log, ad the total umber of colors used i P t or ay coverig path Q i is at most U t, out of the (1+θ 2 ) available. Further ote that because of our order of exposure, there is a set T of size at most 3log such that v s D 2 -out-eighbors are uiformly distributed over V 1 \T. This is because we have exposed whether vu was a D 2 -edge, for the predecessor u of v alog P t, which elimiates oe vertex, ad we may also have already exposed the first half of v s D 2 -out-eighbors i a prior roud, which could cosume up to 1 2 5log vertices. Therefore, the chace that a give out-eighbor exposure is successful (i.e., lads iside U t, via oe of the U t uused colors), is at least ( ) ( ) U t \T V 1 \T U t (1+θ 2 ) 2 3 log 1 (1 o(1)) 2 3 log 1 1 > (1+θ 2 ) 5(log) 2/3. We coclude that the chace that all k 1 42 log fail is at most ( ) log (1+o(1)) 1 5(log) 2/3 < e log. Sice we will ot perform this experimet more tha twice for each of the vertices, liearity of expectatio ad Markov s iequality imply that whp, the fial total umber of red vertices is at most e log, as desired. 3.5 Proof of Lemma 2.5 Atthispoit, wehavearaibowg 2 -pathp oflegth 3 log, whichisdisjoitfromthepaths Q i which cover S. Recall from (6) ad (7) that we defied L = max { 10e 40/(ǫ 3θ 3 ), 7 θ 1 } < 10 log. Split P ito r = L segmets of legth L, as i Figure 1. If is ot divisible by L, we may discard the remaider of P, because L < 10 log. 15

16 Partitio the 2r edpoits ito two sets A 1 B 1 so that each segmet has oe edpoit i each set, but there are o vertices a A 1 ad b B 1 which are cosecutive alog P. By possibly discardig the fial segmet (which will oly cost a additioal L < 10 log), we may esure that the iitial ad fial edpoits are both i A 1. The reaso for our uusual partitio is as follows. I our costructio thus far, we already eeded to expose the locatios of some D1 -edges, sice they had priority over the D 2. I certai locatios, we have revealed that there are o D1 -edges. I particular, betwee every cosecutive pair of vertices u,v o the path P, we foud a D 2 -edge, ad cofirmed the absece of ay D1 -edges. Fortuately, our costructio did ot expose ay D1 -edges betwee o-cosecutive vertices of the path P. I particular, if we ow wished, for ay vertex a A 1, we could expose the umber N of its D 1 -out-eighbors that lie i B 1 ; the, the distributio of these N out-eighbors would be uiform over B 1. This uiformity is crucial, ad would ot hold, for example, if some vertex of B 1 were cosecutive with a alog P. TheproofofLemma2.5breaksitothefollowigsteps. Recallthatd + 1 (v;t)deotestheumber of D 1 -edges from a vertex v to a subset T of vertices. Say that v is T-good if d + 1 (v;t) ǫ 1θ 1 181L log; call it T-bad otherwise. Step 1. For every vertex v P \ (A 1 B 1 ), as well as for the iitial ad fial edpoits of P, expose the value of d + 1 (v;b 1). We show that whp, the iitial ad fial edpoits of P are B 1 -good, ad at most e log vertices of P \(A 1 B 1 ) are B 1 -bad. Step 2. We will describe below a procedure that will allow us to absorb all remaiig vertices ad coverig paths ito a ew system of segmets, usig G 3 -edges that are aliged with B 1 -good vertices. (See Figure 2.) This removes some segmet edpoits, while addig other ew edpoits. Let A 2 B 2 be the ew partitio of edpoits. Crucially, B 2 = B 1, while A 2 = A 1 by losig up to 2 3 log vertices, ad the addig back the same umber. Importatly, every ew vertex i A 2 \A 1 will be B 2 -good. Step 3. We will the show that the system of segmets ca be grouped ito several blocks of cosecutive segmets, i the sese that betwee successive segmets i the same block, there is a origial edge of P. (See Figure 3.) Also, the iitial ad fial edpoits of each block are always of type A, ad are all B 2 -good, ad each block will cotai at least 3 log segmets. L 2 Step 4. For every vertex a A 2, expose the value of d + 1 (a;b 2), ad for every vertex b B 2, expose the value of d + 1 (b;a 2). We show that whp, at most e log vertices of B 2 are A 2 -bad, ad every block cotais a strig of four cosecutive segmets, each of whose A 2 -edpoits ad B 2 -edpoits are B 2 -good ad A 2 -good, respectively. Step 5. For each cosecutive pair of segmets alog the same block (from Step 3) which has either a A 2 -edpoit which is B 2 -bad or a B 2 -edpoit which is A 2 -bad, merge them, together A 1 B 1 B 1 A 1 A 1 B 1 B 1 A 1 B 1 A 1 Figure 1: The log path P, divided ito cosecutive segmets of legth L. Edpoits of successive segmets are adjacet via origial edges of P. The set of segmet edpoits has bee partitioed ito A 1 B 1. Note that edpoits that are adjacet via a edge of P are always assiged to the same set. 16

17 with a eighborig segmet i order to maitai parity betwee A s ad B s. (See Figure 4; the fial claim of Step 4 esures there is a way to accomplish this without ruig out of segmets.) LetA 3 B 3 bethefialpartitioofsegmetedpoitsafterthemergig. Weshow that whp, all a A 3 have d + 1 (a;b 3) ǫ 1θ 1 200L log, ad all b B 3 have d + 1 (b;a 3) ǫ 1θ 1 200L log. Furthermore, if we were to expose the D 1 -edges betwee A 3 ad B 3, the each vertex a A 3 would idepedetly sample d + 1 (a;b 3) uiformly radom eighbors i B 3, ad similarly for b B 3. This will complete the proof because the fial umber of segmets is A 3 = (1 o(1)) L Step 1 By costructio, B 1 = (1 o(1)) L. Now cosider a arbitrary vertex v P \B 1. We have oly exposedtheumericvalueofd 1 (v)thusfariourcostructio, adotwherethed 1-out-eighbors are. So let us ow expose the umeric value of d + 1 (v;b 1), but agai, ot precisely where the outedpoits are. As we observed i the begiig of Sectio 3.4, we have d 1 (v) ǫ 1θ 1 20 log 3. Our work i the previous sectio cosumes up to oe D 1 -out-edge at each vertex v P, whe we reveal whether vw D1 i the ) third step of the algorithm. Therefore, d+ 1 (v;b 1) stochastically domiates log 4, L. Hece we ca use Lemma 3.1 to boud the probability that d + 1 (v;b 1) is Bi( ǫ1 θ 1 20 too small. [ P d + 1 (v;b 1) < ] 20 log ] L ǫ1θ 1 [ P d + 1 (v;b 1) < ǫ 1θ 1 181L log < e L ǫ1 θ 1 20 log = o(e ǫ 1 θ 1 40L log ) = o(e (log)8/9 ), sice ǫ 1,θ 1 = Ω ( 1 loglog ) ad L < 10 log. The expected umber of such vertices i P is at most times this probability. Applyig Markov s iequality, we coclude that whp, the umber of B 1 -bad vertices i P \(A 1 B 1 ) is at most e log, with room to spare. This also shows that the iitial ad fial edpoits of P are B 1 -good whp Step 2 At this poit, our etire vertex set is partitioed as follows. We have a collectio of raibow segmets I 1,...,I r, each of legth exactly L. These already cosume at least 3 log vertices. Sice ) we discarded the remaider of P, as well as possibly the fial segmet, we have r L( 1 3 log 2. A separate collectio of raibow paths Q1,...,Q s covers all vertices of S. There are also some remaiig vertices. I this sectio, we will use G 3 -edges to absorb the latter two classes ito the raibow segmets. Sice we will ot use ay further G 2 -edges, but edges from D2 take precedece over those from D 3, we also ow expose all edges i D2. Lemma 3.3 esures that whp, o vertex is icidet to more tha 5log edges of D2. Coditio o this outcome. Note that by costructio, we have ot exposed the locatios of ay D3 -edges betwee vertices outside S, although vertices outside S may have up to three exposed D 3 -eighbors located i S. 17

18 B 1-good B 1-good A 1 B 1 B 1 A 1 A 1 B 1 B 1 A 1 A 1 B 1 B 1 A 1 G 3 G 3 Coverig path for S Shorteed itervals Merged two cosecutive itervals A 1 A 2 B 1 B 1 A 1 A 1 B 1 B 1 A 2 B 1 B 1 A 1 New iterval Figure 2: A coverig path of S is absorbed ito the system of segmets usig G 3-edges. Note that the resultig edpoit partitio still has oe A-edpoit ad oe B-edpoit i every segmet. Importatly, the directio of the splicig is such that all ew edpoits are of type-a, as idicated by the A 2-vertices. This is why we cotiue the ew segmet rightward, through to the ext B-edpoit. A 1 B 1 B 1 A 1 A 1 x A 2 B 1 B 1 A 1 B 1 A 2 U A 2 A 2 B 1 B 1 A 1 A 1 B 1 B 1 A 1 B 1 A 2 * y A 2 B 1 B 1 A 1 A 1 B 1 B 1 A 1 B 1 A 1 Figure 3: Evolutio of block partitio durig absorptio. Each horizotal row represets a origial block, withi which the successive segmets have their edpoits coected by edges of the origial path P. The vertical gray path from x to y represets the absorptio of a ew vertex ito the collectio of segmets, ivolvig two differet blocks. This operatio cuts the two edges betwee x,y ad their adjacet A 2-vertices, ad adds back the P-edge marked by the asterisk. Afterward, the segmets ca be re-partitioed ito ew blocks (see the gray dotted lies), with all iitial ad fial edpoits i each block of type-a, ad B 1-good. A -bad 2 Merged cosecutive itervals A 2 B 2 B 2 A 2 A 2 B 2 A 2 B 2 Figure 4: Origial edges of P are used to merge cosecutive segmets i the same block, so that a bad edpoit ca be elimiated. 18

19 We ow use a simple greedy algorithm to absorb all residual paths ad vertices ito our collectio of segmets. I each step, we fid a pair of G 3 -edges likig either a ew Q i or a ew vertex to two distict segmets I x ad I y, usig two ew colors from C 3. We will esure that throughout the process, all segmets I w used i this way are separated by at least 3 log full segmets I L 2 z alog P. The specific procedure is as follows. Suppose we have already liked i t paths or vertices, ad are cosiderig the ext path or vertex to lik i. Suppose it is a path Q i (the vertex case ca be treated i a aalogous way). Let u,v be the edpoits of Q i. We eed to fid vertices x,y i distict segmets I x ad I y such that (i) accordig to P, I x ad I y are separated by at least 3 log L 2 full segmets from each other, ad from all I z previously used i this stage, (ii) x ad y are separated from the edpoits of I x ad I y by at least two edges of P, ad (iii) if x I x is the vertex adjacet to x i the directio of the B 1 -edpoit of I x, ad y I y is the vertex adjacet to y i the directio of the B 1 -edpoit of I y, the both x ad y are B 1 -good. We choose the directio of the B 1 -edpoit because x ad y will become the ew A 2 -edpoits of shorteed segmets; see Figure 2 for a illustratio. So, let F be the set of vertices i the segmets which fail properties (ii) or (iii). By Step 1, the domiat term arises from the edpoits because e log 10 log < L, so F < 5 L. Also let T be the set of vertices cotaied i segmets that are at least 3 log full segmets away from L 2 ay segmets which have previously bee touched by this algorithm. Sice we observed at the begiig of this sectio that the total umber of segmets was at least L( 1 ) 3 log 2, we have ( ( 1 T ( L 3 ) 2 ) ) 2 3 log log L 2 (2t) L > 4.5 L, sice t < log ad L < 10 log. Let us ow fid a ewly-colored G 3 -edge from u to a vertex of T \F. Note that the umber of vertices outside T \F is at most 9.5 L. We have ot yet exposed the specific locatios of the D 3 -eighbors of u, but oly kow (sice u S) that d + 3 (u) ǫ 3θ 3 20 log, ad up to three of those D 3-out-eighbors lie withi S. Cosider what happes whe we expose the locatio w of oe of u s D 3 -out-eighbors which is outside S. This will produce a useful G 3 -edge uw if (i) w lads i T \B, (ii) either uw or wu appeared i D1 or D 2, (iii) wu does ot appear i D3, ad (iv) the color of the edge is ew. Let us boud the probability that w fails ay of these properties. We may cosider (i)-(iii) together, sice we showed that at most 9.5 L vertices were outside T \B, ad we coditioed o u beig icidet to at most 5log edges of D2. After samplig the locatio of w, we expose whether uw or wu appear i D1 ; by the same argumet as used i Lemma 3.10, the probability of each is at most 7log = o ( ) 1 L. So, the probability of failig either (i) or (ii) is at most 9.6 L. We coditioed at the begiig of Sectio 3.4 o d + 3 (w) 5log, so whe we expose whether wu is i D3, we agai fail oly with probability at most 7log = o( 1 L ). Fially, whe we expose the color of the ew edge, we kow that it will be i C 3, so the probability that it is a previously used color is at most (2t) 1 θ 3 < 2 3 log 1 θ 3 = 2 θ 3 3 log = o ( ) 1. L Therefore, the probability that all of the at least ǫ 3θ 3 20 log 3 D 3-out-edges of u fail is at most ( ) 10 ǫ 3 θ 3 20 log 3 ( ) L 3 ( = ǫ 3 θ 3 20 log L L 3 10 L 10 10) 2, 19

20 where we used the defiitio of L i (6) for the fial boud. A similar calculatio works for v, ad for the separate case whe we icorporate a ew vertex ito the segmets. Therefore, takig a uio boud over the o() iteratios i likig vertices ad paths, we coclude that our procedure completes successfully whp Step 3 Step 1 established that whp, the iitial ad fial vertices of the log path P are both B 1 -good, so the origial system of segmets ca be arraged as a sigle block, with successive segmets liked by edges of P. We ow prove by iductio that after the absorptio of each path or vertex i Step 2, the collectio of segmets ca be re-partitioed ito blocks of segmets, such that withi each block, cosecutive segmets have their edpoits liked via P, ad the iitial ad fial edpoits i each block are of type-a, ad B 1 -good. There are two cases, depedig o whether the absorptio ivolves two segmets i the same block (as i Figure 2), or i differet blocks (as i Figure 3). If the segmets are i the same block, the we ca easily divide that block ito two blocks satisfyig the coditio. Ideed, i Figure 2, oe of the ew blocks is the strig of segmets betwee the vertices idicated by A 2 i the diagram, ad the other ew block is the complemet. This works because withi each of the two ew blocks, every edge betwee successive segmets was a edge betwee successive segmets of the origial block, hece i P. Also, of the four iitial/fial edpoits amog the two ew blocks, two of them were the iitial/fial edpoits of the origial block, ad the other two were idetified as B 1 -good vertices, ow i A 2. Therefore, the ew block partitio satisfies the requiremets. O the other had, if the absorptio ivolves two segmets from differet blocks, the oe ca re-partitio the two blocks ito three ew blocks, as illustrated i Figure 3. A similar aalysis to above the completes the argumet. Clearly, sice the itervals chose for the splicig are separated by at least 3 log segmets, it also easily follows that the resultig blocks cotai at least L 2 that may segmets as well Step 4 We have ot yet revealed aythig about the D 1 -out-eighbors of ay vertices i B 2 = B 1 ; the oly thig we kow is that they had d + 1 ǫ 1θ 1 20 log. For each vertex b B 1, let us ow expose the umeric value of d + 1 (b;a 2), but agai, ot precisely where the out-edpoits are. Sice our absorptio procedure maitaied A 2 = A 1 = (1 o(1)) L, the same argumet that we used for Step 1 ow establishes the boud for Step 4, with plety of room to spare. It remais to show that every block cotais a strig of 4 cosecutive segmets, all 8 of whose edpoits are good. Cosider a arbitrary block. Step 3 esures that this block has at least 3 log segmets, which costitute at least 3 log disjoit strigs of 4 cosecutive segmets. By L 2 4L 2 the argumet i Step 1, the probability that a particular edpoit is bad is o(e (log)8/9 ), so the probability that a particular strig of 4 segmets has a bad edpoit is at most 8 times that boud, which is also o(e (log)8/9 ). Therefore, the probability that all of these disjoit strigs are bad i this block is at most 3 ( log e (log)8/9) 4L 2 < e (log)1.02 = o( 1 ), 20