Embedding nearly-spanning bounded degree trees
aa r X i v : . [ m a t h . C O ] J un Embedding nearly-spanning bounded degree trees
Noga Alon ∗ Michael Krivelevich † Benny Sudakov ‡ Abstract
We derive a sufficient condition for a sparse graph G on n vertices to contain a copy of a tree T of maximum degree at most d on (1 − ǫ ) n vertices, in terms of the expansion properties of G .As a result we show that for fixed d ≥ < ǫ <
1, there exists a constant c = c ( d, ǫ ) suchthat a random graph G ( n, c/n ) contains almost surely a copy of every tree T on (1 − ǫ ) n verticeswith maximum degree at most d . We also prove that if an ( n, D, λ )-graph G (i.e., a D -regulargraph on n vertices all of whose eigenvalues, except the first one, are at most λ in their absolutevalues) has large enough spectral gap D/λ as a function of d and ǫ , then G has a copy of everytree T as above. In this paper we obtain a sufficient condition for a sparse graph G to contain a copy of every nearly-spanning tree T of bounded maximum degree, in terms of the expansion properties of G . Therestriction on the degree of T comes naturally from the fact that we consider graphs of constantdegree. Two important examples where our condition applies are random graphs and graphs with alarge spectral gap.The random graph G ( n, p ) denotes the probability space whose points are graphs on a fixed set of n vertices, where each pair of vertices forms an edge, randomly and independently, with probability p .We say that the random graph G ( n, p ) possesses a graph property P almost surely , or a.s. for short,if the probability that G ( n, p ) satisfies P tends to 1 as the number of vertices n tends to infinity. ∗ Schools of Mathematics and Computer Science, Raymond and Beverly Sackler Faculty of Exact Sciences, TelAviv University, Tel Aviv 69978, Israel and IAS, Princeton, NJ 08540, USA. Email: [email protected]. Researchsupported in part by a USA-Israeli BSF grant, by NSF grant CCR-0324906, by a Wolfensohn fund and by the Stateof New Jersey. † Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, TelAviv 69978, Israel. E-mail: [email protected]. Research supported in part by USA-Israel BSF Grant 2002-133,and by grants 64/01 and 526/05 from the Israel Science Foundation. ‡ Department of Mathematics, Princeton University, Princeton, NJ 08544. E-mail: [email protected] supported in part by NSF CAREER award DMS-0546523, NSF grant DMS-0355497, USA-Israeli BSF grant,and by an Alfred P. Sloan fellowship. G ( n, c/n ) a.s. contains a path of length at least (1 − α ( c )) n , where α ( c ) is a constant smaller thanone for all c > c →∞ α ( c ) = 0. This conjecture was proved by Ajtai, Koml´os and Szemer´edi[1] and, in a slightly weaker form, by Fernandez de la Vega [11]. These results were significantlyimproved by Bollob´as [5], who showed that α ( c ) decreases exponentially in c . Finally Frieze [16]determined the correct speed of convergence of α ( c ) to zero and proved that α ( c ) = (1 + o (1)) ce − c .The question of existence of large trees of bounded degree other than paths in sparse random graphswas studied by Fernandez de la Vega in [12]. He proved that there exist two constants a > a > T of order n/a with maximum degree at most d the random graph G ( n, c/n ) with c = a d almost surely contains T . The constant a in this result is rather large andallows to embed only trees that occupy a small proportion of the random graph. Also, observe thatFernandez de la Vega’s result gives the almost sure existence of a fixed tree T , and not of all suchtrees simultaneously.Our first theorem improves the result of Fernandez de la Vega and generalizes the above mentionedresults on the existence of long paths. It shows that the sparse random graph contains almost surelyevery nearly-spanning tree of bounded degree. Theorem 1.1
Let d ≥ , < ε < / and let c ≥ d log d log (2 /ε ) ε . Then almost surely the random graph G ( n, c/n ) contains every tree of maximum degree at most d on (1 − ε ) n vertices. Results guaranteeing the existence of a long path in a sparse graph can be obtained in a more generalsituation when the host graph has certain expansion properties. Given a graph G = ( V, E ) and asubset X ⊂ V let N G ( X ) denote the set of all neighbors of vertices of X in G . Using his celebratedrotation-extension technique, P´osa [18] proved that if for every X in G with | X | ≤ k we have that | N G ( X ) \ X | ≥ | X | −
1, then G contains a path of length 3 k −
2. A remarkable generalization of thisresult from paths to trees of bounded degree was obtained by Friedman and Pippenger [15]. Theyproved that if | N G ( X ) | ≥ ( d + 1) | X | for every set X in G with | X | ≤ k −
2, then G contains everytree with k vertices and maximum degree at most d . Note that this result allows to embed only treeswhose size is relatively small compared to the size of G . What if we want to embed trees which arenearly-spanning? It turns out that a slightly stronger expansion property, based on the spectral gapcondition, is already enough to attain this goal.For a graph G let λ ≥ λ ≥ . . . ≥ λ n be the eigenvalues of its adjacency matrix. The quantity λ ( G ) = max i ≥ | λ i | is called the second eigenvalue of G . A graph G = ( V, E ) is called an ( n, D, λ ) -graph if it is D -regular, has n vertices and the second eigenvalue of G is at most λ . It is well known2see, e.g., [3] for more details) that if λ is much smaller than the degree D , then G has strongexpansion properties, so the ratio D/λ could serve as some kind of measure of expansion of G . Ournext result shows that an ( n, D, λ )-graph G with large enough spectral gap D/λ contains a copy ofevery nearly-spanning tree with bounded degree.
Theorem 1.2
Let d ≥ , < ε < / and let G be an ( n, D, l ) -graph such that Dλ ≥ d / log(2 /ε ) ε . Then G contains a copy of every tree T with (1 − ε ) n vertices and with maximum degree at most d . Our main results are tightly connected to the notion of universal graphs. For a family H of graphs,a graph G is H -universal if G contains every member of H as a (not necessarily induced) subgraph.The construction of sparse universal graphs for various families arises in several fields such as VLSIcircuit design, data representation and parallel computing (see, e.g., the introduction of [2] for ashort survey and relevant references). Our two main results show that sparse random graphs andpseudo-random graphs on n vertices are universal graphs for the family of bounded-degree trees on(1 − ε ) n vertices. Quite an extensive research exists on universal graphs for trees [4], [7], [8], [9], [10],[15]. The most interesting result is that of Bhatt et al. who showed in [4] that there exists a universalgraph G on n vertices for the family of trees on n vertices with maximum degree d , whose maximumdegree is bounded by a function of d . It is instructive to compare our results with those of [4]: theysucceed in embedding spanning trees as opposed to nearly-spanning in our case; on the other hand,their universal graph is a concrete carefully constructed graph that has very dense pieces locally,while we provide a very large family of universal graphs possessing many additional properties thatcan be useful for obtaining further results on universal graphs.The results of Theorem 1.1 and 1.2 can be deduced from a more general statement which we presentnext. We need the following definition. Definition 1.3
Given two positive numbers c and α < , a graph G = ( V, E ) is called an ( α, c ) -expander if every subset of vertices X ⊂ V ( G ) of size | X | ≤ α | V ( G ) | satisfies: | N G ( X ) | ≥ c | X | . Theorem 1.4
Let d ≥ , < ε < / . Let G = ( V, E ) a graph on n vertices of minimum degree δ = δ ( G ) and maximum degree ∆ = ∆( G ) . Let n, δ, ∆ satisfy:1. (order of graph is large enough) n ≥ d log(2 /ε ) ε ;3 . (maximum degree is not too large compared to the minimum degree) ∆ ≤ K e δ/ (8 K ) − where K = 20 d log(2 /ε ) ε .
3. (local expansion) Every induced subgraph G of G with minimum degree at least εδ d log(2 /ε ) isa ( d +2 , d + 1) -expander.Then G contains a copy of every tree T on at most (1 − ε ) n vertices of maximum degree at most d . The rest of this paper is organized as follows. In the next two sections we show how Theorem 1.4 canbe used to embed nearly-spanning trees of bounded degree in random and pseudo-random graphs.We present the proof of Theorem 1.2 first, since it is short and less technical, and then prove Theorem1.1. In Section 4 we describe the plan of the proof of Theorem 1.4 and discuss some technical toolsneeded to fulfill this plan. The proof of this theorem appears in Section 5. The last section of thepaper contains several concluding remarks and open problems.Throughout the paper we make no attempts to optimize the absolute constants. To simplify thepresentation, we often omit floor and ceiling signs whenever these are not crucial. Throughout thepaper, log denotes logarithm in the natural base e . In this section we prove Theorem 1.2. First we need the following lemma that shows that an ( n, D, l )-graph has the local expansion property required by Condition 3 of Theorem 1.4.
Lemma 2.1
Let d ≥ . Let G = ( V, E ) be an ( n, D, l ) -graph. Denote D = 2 l ( d + 1) √ d . Then every induced subgraph G of G of minimum degree at least D is a (cid:0) d +2 , d + 1 (cid:1) -expander. Proof.
We will use the following well known estimate on the edge distribution of an ( n, D, l )-graph G (see, e.g., [3], Corollary 9.2.5). For every two (not necessarily disjoint) subsets B, C ⊆ V ,let e ( B, C ) denote the number of ordered pairs ( u, v ) with u ∈ B, v ∈ C such that uv is an edge.Note that if u, v ∈ B ∩ C , then the edge uv contributes 2 to e ( B, C ). In this notation, (cid:12)(cid:12)(cid:12)(cid:12) e ( B, C ) − | B || C | Dn (cid:12)(cid:12)(cid:12)(cid:12) ≤ l p | B || C | . Let U be a subset of vertices of G such that the induced subgraph G = G [ U ] has minimumdegree at least D . Suppose that the claim is false. Then there exists a subset X ⊂ U of size | X | = t ≤ | U | / (2 d + 2) satisfying | N G ( X ) | < ( d + 1) | X | .4y the above estimate with B = X and C = N G ( X ) we have: D t ≤ e ( B, C ) ≤ t ( d + 1) tDn + lt √ d + 1and therefore tn ≥ D ( d + 1) D − l √ d + 1 D . (1)Also, note that there are no edges of G from X to Y = U − ( X ∪ N G ( X )) as G is an inducedsubgraph of G . From t = | X | ≤ | U | / (2 d + 2) and | N G ( X ) | ≤ ( d + 1) t it follows that | Y | ≥ dt . Thus0 = e ( X, Y ) ≥ t ( dt ) Dn − l p t ( dt ) , implying tn ≤ l √ dD . (2)Comparing (1) and (2) we obtain D ( d + 1) D − l √ d + 1 D ≤ l √ dD . Plugging in the definition of D we derive a contradiction. ✷ Proof of Theorem 1.2.
Since every graph with minimal degree k contains all the trees on k vertices we can assume that D ≤ (1 − ε ) n . Let A be the adjacency matrix of G . The trace of A equals the number of ones in A , which is exactly 2 | E ( G ) | = nD . We thus obtain that nD = T r ( A ) = n X i =1 λ i ≤ D + ( n − λ and therefore λ ≥ D ( n − D ) n − ≥ εD . This together with our assumption on D/λ implies n ≥ D ≥ ε (cid:18) Dλ (cid:19) ≥ d log (2 /ε ) ε . Since ∆( G ) = (. G ) = D , from this inequality it follows that G satisfies Conditions 1, 2 of Theorem1.4. Finally since εD d log(2 /ε ) ≥ d + 1) l √ d we can conclude using Lemma 2.1 that G also satisfies the last condition of Theorem 1.4. Thus G contains every tree of size (1 − ε ) n with maximum degree at most d . ✷ Embedding in random graphs
To prove Theorem 1.1 we first need to show that a sparse random graph contains a.s. a nearlyspanning subgraph with good local expansion properties.
Lemma 3.1
For every integer d ≥ , real < θ < / and D ≥ θ − the random graph G (cid:0) n, Dn (cid:1) almost surely contains a subgraph G ∗ having the following properties:1. | V ( G ∗ ) | ≥ (1 − θ ) n ;2. D ≤ d G ∗ ( v ) ≤ D for every v ∈ V ( G ∗ ) ;3. every induced subgraph G of G ∗ of minimum degree at least D = 100 d log D is a (cid:0) d +2 , d + 1 (cid:1) -expander. The following statement contains a few easy facts about random graphs.
Proposition 3.2
Let G ( n, p ) be a random graph with np > , then almost surely ( i ) The number of edges between any two disjoint subsets of vertices A, | A | = a and B, | B | = b with abp ≥ n is at least abp/ and at most abp/ . ( ii ) Every subset of vertices of size a ≤ n/ spans less than anp/ edges. Proof. (i) Since the number of edges between A and B is a binomially distributed randomvariable with parameters ab and p , it follows by the standard Chernoff-type estimates (see, e.g., [3])that (denoting t = abp/ P h e ( A, B ) − abp < − t i ≤ e − t abp = e − abp/ and P h e ( A, B ) − abp > t i ≤ e − t abp + t abp )2 = e − abp/ . Using that abp ≥ n we can bound the probability that there are sets A, B with | e ( A, B ) − abp | >abp/ n · n · (cid:0) e − n (cid:1) = o (1).Since np/ ≥
10 and n/a ≥
4, the probability that there is a subset of size a which violates theassertion (ii) is at most P a ≤ (cid:18) na (cid:19)(cid:18) a / anp/ (cid:19) p anp/ ≤ (cid:18) ena (cid:16) eanp (cid:17) np/ p np/ (cid:19) a = e np/ ( n/a ) np/ − ! a ≤ (cid:18) e ( n/a ) (cid:19) a . It is easy to see that P a ≪ n − for all a ≤ n/ P a P a = o (1). ✷ roof of Lemma 3.1. Let G = G ( n, p ) be a random graph with p = Dn and let X be the setof θn/ G . By Part (ii) of Proposition 3.2, a.s. this set spans less than | X | np/ D | X | edges. Also, since Dn | X | ( n − | X | ) ≥ Dθ ( n/ ≥ n , Part (i) of this propositionimplies that a.s. the number of edges between X and V ( G ) − X is at most 3 | X | np/ D | X | .Therefore the sum of the degrees of the vertices in X is bounded by 10 D | X | and hence there is avertex in X with degree at most 10 D . By definition of X , this implies that there are at most θn/ G with degree larger than 10 D . Delete these vertices and denote the remaining graphby G ′ . Next as long as G ′ contains a vertex v of degree less than D , delete it. If we deleted morethan θn/ Y and V ( G ′ ) − Y such that | Y | = θn/ | V ( G ′ ) − Y | ≥ (1 − θ ) n ≥ n/ D | Y | ≤ p | Y || V ( G ′ ) − Y | / Dn | Y || V ( G ′ ) − Y | ≥ θDn ≥ n , again by Part (i) of the previous statementthis a.s. does not happen. Denote the resulting graph by G ∗ . Then it satisfies the first two conditionsof the lemma and it remains to verify the third condition.Suppose to the contrary that G ∗ contains a subset of vertices U such that the induced subgraph G = G ∗ [ U ] has minimum degree at least D = 100 d log D and is not a (cid:0) d +2 , d + 1 (cid:1) -expander. Thenthere exists a set X ⊂ U of size | X | = t such that the set C = N G ( X ) has size at most ( d + 1) t andthere are at least D | X | / dt log D edges with an end in X and another end in C . If t ≤ log DD n ,then the probability that G ( n, p ) contains such sets is at most P t ≤ (cid:18) nt (cid:19)(cid:18) n ( d + 1) t (cid:19)(cid:18) t ( d + 1) t dt log D (cid:19) p dt log D ≤ "(cid:16) ent (cid:17) (cid:18) en ( d + 1) t (cid:19) d +1 (cid:18) e ( d + 1) tp d log D (cid:19) d log D t ≤ " e (cid:16) nt (cid:17) d (cid:18) e · Dtn log D (cid:19) d log D t = " e (cid:16) e (cid:17) d log D (cid:18) D log D (cid:19) d (cid:18) Dtn log D (cid:19) d log D − d t < " e − d log D +2 d log D +1 (cid:18) tn log D/D (cid:19) d log D t ≤ " D − d (cid:18) tn log D/D (cid:19) d log D t . Checking separately two cases t < log n and log n ≤ t < log DD n it is easy to see that in both P t ≪ n − . If t ≥ log DD n we apply a different argument. Note that there are no edges of G ( n, p ) from X to C = U − ( X ∪ N G ( X )) since G is an induced subgraph. From t = | X | ≤ | U | / (2 d + 2) and | N G ( X ) | ≤ ( d + 1) t it follows that | C | ≥ dt and therefore the probability of such event in G ( n, p ) is7t most P t ≤ (cid:18) nt (cid:19)(cid:18) ndt (cid:19) (1 − p ) dt ≤ (cid:20) ent · (cid:16) endt (cid:17) d e − pdt (cid:21) t ≤ (cid:20)(cid:16) ent (cid:17) d e − pdt (cid:21) t = (cid:20)(cid:16) ent (cid:17) · e − pt (cid:21) dt ≤ "(cid:18) enn log D/D (cid:19) · e − Dn · log DD n dt ≤ (cid:0) D D − (cid:1) dt = o ( n − ) . Thus the probability that G ∗ fails to satisfy the third condition is at most P nt =1 P t = o (1). ✷ Proof of Theorem 1.1.
Let d ≥
2, 0 < ε < / c satisfy the assumption of Theorem1.1. Set θ = 0 . ε , D = c/ ε = ε − θ − θ ≥ . ε . Then by Lemma 3.1 G ( n, c/n ) almostsurely contains a subgraph G ∗ of order n ≥ (1 − θ ) n such that D ≤ (. G ∗ ) ≤ ∆( G ∗ ) ≤ D andevery induced subgraph of G ∗ with minimum degree at least 100 d log D is an (cid:0) d +2 , d + 1 (cid:1) -expander.Using that ∆( G ∗ ) ≤ G ∗ ) and n ≥ (. G ∗ ) ≥ D ≥ d log d log (2 /ε )4 ε > d log(2 /ε ) ε we conclude that G ∗ satisfies Conditions 1 and 2 of Theorem 1.4 (with ε ). To verify the thirdcondition it is enough to check that the assumptions in Theorem 1.1 imply that100 d log D ≤ ε D d log(2 /ε ) . Note that one can simply substitute the lower bound for D in the above expression, since x/ log x is an increasing function for x >
3. Therefore by Theorem 1.4, G ∗ contains every tree of size(1 − ε ) n ≥ (1 − ε )(1 − θ ) n = (1 − ε ) n with maximum degree at most d . ✷ To prove Theorem 1.4 we will use the following framework. Given a tree T , we first cut it intosubtrees T , T , . . . , T s of carefully chosen sizes, so that the number s of these subtrees satisfies s ≤ d log(2 /ε ), and each subtree T i is connected by a unique edge to the union of all previoussubtrees. The subtrees T i will be embedded sequentially in order, starting from T .We then choose s pairwise disjoint sets of vertices S , S , . . . , S s whose total size is at most εn/ S i . The set S i will be used onlywhen embedding the subtree T i , and will not be touched before that step. (It can be used later, butwe will not do it here, as it complicates matters and does not improve the estimates in any essential8ay). During the embedding process we maintain a set R of at most s vertices, which will consist ofall roots of the trees T i that still have to be embedded, and will not contain any vertex of the sets S i .At the i -th step we are to embed the tree T i starting from a given root x i ∈ R . (At the first step aroot is chosen arbitrarily.) Suppose that the current set of unused vertices of G is V i − . We take anarbitrary subset U i of size | U i | = Θ( | V ( T i ) | d ) which contains the vertex x i that will be the root of T i (but contains no other members of R ), contains the set S i , and contains no member of S j for j = i .Note that as each vertex has many neighbors in S i , the minimum degree in the induced subgraph of G on U i is large.Since the minimum degree of G [ U i ] is large enough, we can use the result of Friedman and Pippengerto embed a copy of T i in U i , rooting it at x i . (We actually need a slightly modified, rooted version,of their result). All vertices of U i unused when embedding T i are recycled, and we thus get V i bydeleting from V i − only the vertices used for embedding T i .The final step of embedding T i is to embed the edges crossing from T i to yet unembedded pieces T j ,with j > i (using vertices of U i ). We then add the endpoints of those edges outside T i to the list R of special vertices, and delete x i from R . Each of the newly added special vertices will serve asa root for embedding the corresponding piece T j . Observe that the number of special vertices is atmost s at any stage of the embedding.The precise technical details are described in what follows. The cornerstone of our proof is the embedding result of Friedman and Pippenger. In fact, we needa slightly stronger version of it – they showed the existence of a tree T , while we need to embed arooted version of T in G starting from a fixed vertex v ∈ V ( G ) as its root. Luckily, a careful readingof [15] reveals that the following holds as well. Theorem 4.1 ([15])
Let T be a tree on k vertices of maximum degree at most d rooted at r . Let H = ( V, E ) be a non-empty graph such that, for every X ⊂ V ( H ) with | X | ≤ k − , | N H ( X ) | ≥ ( d + 1) | X | . Let further v ∈ V ( H ) be an arbitrary vertex of H . Then H contains a copy of T , rooted at v . Proposition 4.2
Let d ≥ and k be positive integers. Let T be a tree on at least k + 1 vertices withmaximum degree at most d . Then there exists an edge e ∈ E ( T ) such that at least one of the twotrees obtained from T by deleting e has at least k and at most ( d − k −
1) + 1 vertices. roof. Choose a leaf r of T arbitrarily and root T at r . For i ≥ L i the set of verticesof T at distance i from r . For a vertex v ∈ V ( T ) let t ( v ) be the number of vertices in the subtree of T rooted at v . Now, let i = max { i : L i contains a vertex v with t ( v ) ≥ k } . As L has only one vertex v with t ( v ) = | V ( T ) | − ≥ k , it follows that i ≥
1. Choose a vertex u ∈ L i such that t ( u ) ≥ k . Then by the definition of i all sons w of u in T satisfy: t ( w ) ≤ k − d −
1, and therefore t ( u ) ≤ ( d − k −
1) + 1.Let now x be the father of u in T . Then e = ( x, u ) is the required edge. ✷ Corollary 4.3
Suppose < ε < / , and let T be an arbitrary tree on (1 − ε ) n vertices, withmaximum degree at most d . Then one can cut T into s subtrees T , T , . . . , T s , so that each tree T i is connected by a unique edge to the union of all trees T j with j < i , and such that for every i > , εn/ P j>i | V ( T j ) | d ≤ | V ( T i ) | ≤ εn/ P j>i | V ( T j ) | d . For i = 1 , the upper bound holds, but the lower bound may fail. Moreover, s ≤ d log(2 /ε ) . Proof.
We choose the trees T i one by one, starting from the last one. By Proposition 4.2 we firstfind a tree T ′ of size at least εn d and at most εn d , and omit it from T . Suppose we have alreadychosen T ′ , T ′ , . . . , T ′ i − such that for every j < iεn/ P r Let numbers K, δ, ∆ satisfy K ∆ e − ( δ/ K )+1 < . hen the following holds. Let H = ( V, E ) be a graph in which δ ≤ d ( v ) ≤ ∆ for each v ∈ V . Then H contains K pairwise disjoint sets of vertices S , S , . . . , S K such that every vertex of H has at least δ K neighbors in each set S i . Proof. This is a simple consequence of the Lov´asz Local Lemma (c.f., e.g., [3], Chapter 5).Color the vertices of H randomly and independently by K colors. For each vertex v and color i ,1 ≤ i ≤ K , let A v,i be the event that v has less than δ K neighbors of color i . By Chernoff’sInequality the probability of each event A v,i is at most e − δ/ (8 K ) . In addition, each event A v,i ismutually independent of all events but the events A u,j where either u = v or u and v have commonneighbors in H . As there are less than K (cid:0) ∆(∆ − 1) + 1 (cid:1) ≤ K ∆ such events, it follows by the LocalLemma that with positive probability none of the events A v,i holds. The desired result follows, byletting S i denote the set of all vertices of color i . ✷ Let G = ( V, E ) be a graph satisfying the assumptions of the theorem. Let T be a tree on at most(1 − ε ) n vertices with maximum degree at most d . By Corollary 4.3 the tree T can be partitioned intosubtrees T , T , . . . , T s satisfying the conditions of the Corollary, where s ≤ d log(2 /ε ) . Choosean arbitrary root for T . For i > 1, the root of T i is the vertex incident with the unique edge thatconnects T i to the union of the previous trees. Put K = 2 s/ε . By Condition 2 in the theorem, andLemma 4.4 there are K pairwise disjoint sets of vertices S i of G such that every vertex of G has atleast δ/ (2 K ) ≥ εδ d log(2 /ε ) neighbors in each set S i . Take the s smallest sets S i and renumber themso that they are denoted by S , S , . . . , S s . Obviously, their total size is at most nsK = εn . We willnot use all the other sets S i , i > s in the rest of the proof.Let x be an arbitrary vertex of G that does not lie in any of the sets S i . Define R = { x } andlet U denote the set of all vertices of G besides those in ∪ j =1 S j . As U contains S , every vertexin the induced subgraph G [ U ] of G on U has degree at least εδ d log(2 /ε ) . Therefore, by Condition3 in Theorem 1.4, and by Theorem 4.1 there is a copy of T in G [ U ] rooted at x . (Note that byCorollary 4.3, the size of T is at most | U | d and hence indeed one can apply here Theorem 4.1.)Moreover, we can in fact embed in U the required tree T together with the edges connecting it tothe trees T j with j > 1. Add the endpoints of these edges to the list R of planned roots for the trees T j , and delete x from R . This completes the embedding of T .Assume that we have already embedded the first i − T r , where each T r has been rooted in thevertex of R specified as its root, and where in step number r the tree T r has been embedded usingno vertices of R besides its root, and no vertices of ∪ j = r S j , we proceed to the i -th step, in which weare to embed the tree T i starting from a given root x i ∈ R . Let U i be the set of all vertices of G thathave not been used for embedding the part of T embedded so far, besides the vertices in R − { x i } and11esides the vertices in ∪ j = i S j . As before, since U i contains S i , every vertex in the induced subgraph G [ U i ] of G on U i has degree at least εδ d log(2 /ε ) . Therefore, by Condition 3 in Theorem 1.4, it is a( d +2 , d + 1)-expander. By Corollary 4.3, the size of T i is at most | U i | / d and the number of edgesconnecting T i to the trees T j for j > i is bounded by s ≤ d log(2 /ε ) ≤ εn d ≤ | U i | d . Therefore the size of T i together with the vertices in T j for j > i that are connected to it is less thana fraction d ≤ d +2) of the size of U i . Therefore, by Theorem 4.1 we can embed T i including theedges connecting it to the trees T j with j > i in U i , rooting it at x i , and add the endpoints of theedges from T i to future T j ’s to R .As this process can be carried out until we finish the embedding of T s , the assertion of the theoremfollows. ✷ • Our lower bound on the edge probability of a random graph in Theorem 1.1 seems far frombeing best possible, and the correct order of magnitude should probably be more similar tothe case of a longest path. Hence it is likely that already when c = O ( d log(1 /ǫ )) the randomgraph G ( n, c/n ) contains a.s. every tree on (1 − ǫ ) n vertices with maximum degree at most d . • Embedding spanning trees of bounded degree in sparse random graphs is an intriguing questionwhich is completely open. In case of the path, this question is very well understood (see, e.g.,Chapter 8 of [6]) and it is known that for p = O (log n/n ) the random graph G ( n, p ) a.s. containsa Hamiltonian path. We believe that a more general result should be true, i.e., such a randomgraph should already contain a.s. every tree on n vertices with maximum degree at most d .Our methods are insufficient to attack this problem. Using Theorem 1.1 we can only provethe following much weaker result. Let T be a tree on n vertices with at least ǫn leaves, thenthere exists a constant a ( ǫ, d ) such that the random graph G (cid:0) n, a log nn (cid:1) a.s. contains T . Hereis a brief sketch of the proof: we split G ( n, p ) into two random graphs G ( n, p ) and G ( n, p ),where 1 − p = (1 − p )(1 − p ), p = Θ(1 /n ), p = Θ(log n/n ). Let T ′ be the tree obtainedfrom T by deleting its leaves. We use Theorem 1.1 to embed a copy of T ′ in G ( n, p ). Thenwe expose the edges of G ( n, p ) between the set of vertices V of G , not occupied by a copy of T ′ , and the rest of the graph, and embed the leaves of T in V using matching-type results. • Besides the model G ( n, p ), another model of random graphs, drawing a lot of attention isthe model of random regular graphs. A random regular graph G n,D is obtained by samplinguniformly at random over the set of all simple D -regular graphs on a fixed set of n vertices. Bythe result of Friedman, Kahn and Szemer´edi [14] the second eigenvalue of G n,D is almost surely12t most O ( √ D ) (see [13] for a more precise result). Therefore our Theorem 1.2 immediatelyimplies that if D = D ( d, ǫ ) is sufficiently large then G n,D a.s. contains every tree on (1 − ǫ ) n vertices with maximum degree d .This result as well as the result of Theorem 1.2 are probably not optimal. We suspect thatsufficiently large spectral gap (as a function of d only) already suffices to guarantee the embed-ding of every spanning tree of bounded degree in a graph G of order n . This is not known evenfor the Hamiltonian path, and the best result in this case, obtained in [17], requires spectralgap of order roughly log n . Acknowledgment. A major part of this work was carried out when the authors were visitingMicrosoft Research at Redmond, WA. We would like to thank the members of the Theory Group atMicrosoft Research for their hospitality and for creating a stimulating research environment. References [1] M. Ajtai, J. Koml´os and E. Szemer´edi, The longest path in a random graph, Combinatorica Proc. th Int. Workshop on Randomizationand Approximation techniques in Computer Science (RANDOM-APPROX 2001) , Berkeley 2001,170–180.[3] N. Alon and J. H. Spencer, The probabilistic method , 2 nd Ed., Wiley, New York, 2000.[4] S. N. Bhatt, F. Chung, F. T. Leighton and A. 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