Rolling backwards can move you forward: on embedding problems in sparse expanders
aa r X i v : . [ m a t h . C O ] J u l Rolling backwards can move you forward: on embeddingproblems in sparse expanders
Nemanja Draganić ∗ Michael Krivelevich † Rajko Nenadov ‡ Abstract
We develop a general embedding method based on the Friedman-Pippenger tree embed-ding technique (1987) and its algorithmic version, essentially due to Aggarwal et al. (1996),enhanced with a roll-back idea allowing to sequentially retrace previously performed em-bedding steps. This proves to be a powerful tool for embedding graphs of large girth intoexpander graphs. As an application of this method, we settle two problems: • For a graph H , we denote by H q the graph obtained from H by subdividing its edgeswith q − k -size-Ramsey number ˆ R k ( H q ) satisfiesˆ R k ( H q ) = O ( qn ) for every bounded degree graph H on n vertices and for q = Ω(log n ),which is optimal up to a constant factor. This settles a conjecture of Pak (2002). • We give a deterministic, polynomial time algorithm for finding vertex-disjoint pathsbetween given pairs of vertices in a strong expander graph. More precisely, let G be an( n, d, λ )-graph with λ = O ( d − ε ), and let P be any collection of at most c n log d log n disjointpairs of vertices in G for some small constant c , such that in the neighborhood of everyvertex in G there are at most d/ P . Then there exists a polynomialtime algorithm which finds vertex-disjoint paths between every pair in P , and eachpath is of the same length ℓ = O (cid:16) log n log d (cid:17) . Both the number of pairs and the length ofthe paths are optimal up to a constant factor; the result answers the offline version ofa question of Alon and Capalbo (2007). Given a graph H from some class of graphs, and a graph G with specific properties, is therea copy of H in G ? In other words, does there exist an embedding of H into G ? This generalquestion is one of the central settings of combinatorics. Embedding questions lie at the heartof many classical problems, in particular problems in graph Ramsey theory and Turán-typeextremal theory.We will consider embedding problems where the host graph G is sparse, i.e. the number ofedges in G is linear in its number of vertices. This is a natural and important setup both fortheoretical and practical reasons, and its potential applicability ranges from problems in ex-tremal combinatorics like Ramsey-type problems, to construction of lean but resilient networksin computer networking.In particular, we will work with sparse expanders — those are sparse graphs in which all sets ofvertices S of (up to) a certain size have a relatively large neighborhood. For a comprehensive ∗ Department of Mathematics, ETH, 8092 Zürich, Switzerland. Email: [email protected] . † School of Mathematical Sciences, Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv 6997801,Israel. Email: [email protected] . Supported in part by USA-Israel BSF grant 2018267, and by ISFgrant 1261/17. ‡ Department of Mathematics, ETH, 8092 Zürich, Switzerland. Email: [email protected] . n, d, λ )-graphs, introduced by Alon. A d -regular graph G on n vertices is an ( n, d, λ )-graph if all of the eigenvalues of its adjacencymatrix, except the largest one, are at most λ in absolute value. One can show that the smaller λ is, the closer the graph resembles a random graph in terms of edge distribution (see Section1.1.1 for some details). A small λ also means that the graph has good expansion properties andwe will use a few such results throughout the paper. For a survey on pseudo-random graphs,see the paper of Krivelevich and Sudakov [42].For our embedding problems, usually the host graph G will be an ( n, d, λ )-graph, for sufficientlysmall λ and a constant d . What kind of subgraphs can we hope to find in such graphs? Onenatural restriction will be that the girth of the graph we embed is Ω(log n ), as there exist( n, d, λ )-graphs with small spectral ratio λ/d and of logarithmic girth, as shown in the seminalpaper of Lubotzky, Phillips and Sarnak [44]. Thus we are normally confined to embedding treesand other graphs with large girth.There is a large body of research devoted to finding (almost-spanning and spanning) boundeddegree trees in sparse expanders and in sparse random graphs. Beck [10] used results aboutlong paths in expanding graphs to argue that one can find monochromatic linear sized paths in2-colored sparse random graphs. Friedman and Pippenger [28] proved an analogous statementfor arbitrary bounded degree trees in sparse expanders, which was improved upon by Haxell,who showed that under similar assumptions one can embed even larger trees into (sparse)expanders. Alon, Krivelevich and Sudakov [5] proved the existence of every almost spanningtree of bounded degree in both sparse random graphs and in appropriate ( n, d, λ )-graphs, laterimproved by Balogh, Csaba and Samotij and Pei [8], and for a resilience version of this result see[9]. Finally, for random graphs G ∼ G ( n, p ) with p = C log nn and for a fixed d , Montgomery [47]recently proved that for large enough C , G typically contains all spanning trees of maximumdegree at most d , resolving an old conjecture of Kahn. For results about finding small minorsof logarithmic girth in sparse expanders, see, e.g. [46, 54].In this paper, we will show two different results related to embedding into sparse expanders— the first one deals with size-Ramsey numbers of logarithmic subdivisions of bounded degreegraphs and resolves a conjecture of Pak from 2002 [48], while the second is concerned withthe classical problem of finding vertex-disjoint paths in graphs, and solves the offline versionof the problem of Alon and Capalbo from 2007 [3]. For each of those problems we developa separate variation of our embedding technique. Both are based on the result of Friedmanand Pippenger [28] about embedding trees in expander graphs vertex by vertex and an idea byDaniel Johannsen [36], which allows us to successively remove vertices from the list of alreadyembedded vertices. This roll-back result turns out to be very powerful for tackling problemsof this sort. One of the variants which we show is algorithmic, and uses ideas by Dellamonicaand Kohayakawa [20], who showed an algorithmic version of the original Friedman-Pippengerembedding result, by reducing it to a certain online matching problem solved by Aggarwal etal. [1]. Given a graph H and an integer k ≥
2, a graph G is said to be k -Ramsey for H if everycoloring of the edges of G with k colors contains a monochromatic copy of H . This notion wasintroduced by Ramsey [50], who proved that for every graph H there exists N ∈ N such that K N is k -Ramsey for H . The smallest such N , denoted by R k ( H ), is called the Ramsey number .2etermining the asymptotic order of R ( K ℓ ) is one of the most important open problems in thisarea [16, 55]. We will be concerned with the related notion of size-Ramsey numbers , introducedby Erdős, Faudree, Rousseau and Schelp [25]. Given a graph H and an integer k ≥
2, the size-Ramsey number ˆ R k ( H ) is the smallest integer m such that there exists a graph G with m edgeswhich is k -Ramsey for H . The existence of the Ramsey number immediately implies the upperbound ˆ R k ( H ) ≤ (cid:0) R k ( H )2 (cid:1) . Other related notions include Folkman numbers, chromatic-Ramseynumbers, degree-Ramsey numbers, etc. We refer the reader to a recent survey by Conlon, Foxand Sudakov [17] for a thorough treatment of the topic.Answering a 100$–question of Erdős [24], Beck [10] showed that paths have linear size-Ramseynumber, that is ˆ R ( P n ) ≤ Cn for an absolute constant C . He also raised the question [11] ofwhether ˆ R ( H ) grows linearly for graphs with bounded maximum degree. This was proven fortrees by Friedman and Pippenger [28] and for cycles by Haxell, Kohayakawa and Łuczak [32].However, the general case was settled in the negative by Rödl and Szemerédi [51], who showedthat there exists a constant c > n there is a graph H with n vertices and maximum degree 3 for which ˆ R ( H ) ≥ n log c n . In the same paper, they conjecturedthat log c n can be improved to n ε for some constant ε >
0, but this remains open. For furtherresults about size-Ramsey numbers, see for example [7, 12, 15, 19, 23, 31, 35, 37, 41, 43].
Since we are far from understanding size-Ramsey numbers of bounded degree graphs in general,one natural step in this direction is to consider subdivisions of those graphs. Given a graph H and a function σ : E ( H ) → N , the σ -subdivision H σ of H is the graph obtained from H byreplacing each edge e ∈ E ( H ) with a path of length σ ( e ) joining the endpoints of e , such thatall these paths are mutually vertex-disjoint (except possibly at the endpoints). In other words,we subdivide each edge σ ( e ) − R k ( H q ) ≤ O ( n /q ),for constant q, k and for all bounded degree graphs H , thus removing a polylogarithmic factorfrom their bound and answering their question. In general, these graphs were considered in thecontext of Ramsey theory by Burr and Erdős [14] and by Alon [2].In Section 3.1 (Theorem 1.2) we show that bounded degree graphs with n vertices such thatevery two vertices of degree ≥ q = Ω(log n ), have linear size-Ramsey numbers(in their order). In fact we prove a stronger result on arbitrary long subdivisions of boundeddegree graphs, answering a conjecture of Pak [48] along the way. He conjectured that longsubdivisions of bounded degree graphs have linear size-Ramsey number. Conjecture 1.1 ([48]) . For every k, D ∈ N there exist C, L > such that if H is a graph with ∆( H ) ≤ D and σ ( e ) = ℓ ≥ L log( v ( H σ )) for all e ∈ E ( H ) then ˆ R k ( H σ ) ≤ Cv ( H σ ) . Pak [48] showed that ˆ R k ( H σ ) = O ( v ( H σ ) log ( v ( H σ ))) and the special case where H is a fixed(small) graph and σ ( e ) grows was resolved by Donadelli, Haxell and Kohayakawa [21].We show that every η -uniform graph on n vertices is k -Ramsey for H σ with v ( H σ ) ≤ αn and σ ( e ) ≥ log n , for some small α >
0. As a typical random graph with n vertices and m = Cn edges is η -uniform, for sufficiently large C , there are an abundance of η -uniform graphs with O ( n ) edges, thus confirming Conjecture 1.1. Definition.
Given 0 < η ≤
1, we say that a graph G with n vertices and density p = e ( G ) / (cid:0) n (cid:1) is η -uniform if for every disjoint subsets U, W ⊆ V ( G ) of size | U | , | W | ≥ ηn , we have (cid:12)(cid:12) e ( U, W ) − | U || W | p (cid:12)(cid:12) ≤ η | U || W | p. Theorem 1.2.
For every k, D ∈ N and for every δ > , there exist η, α, C > , such thatthe following holds for every η -uniform graph G with n vertices and m ≥ Cn edges: every k -edge-coloring of G contains a monochromatic copy of every graph H σ , where H is a graph withmaximum degree at most D , v ( H σ ) ≤ αn and σ ( e ) ≥ δ log n for every e ∈ E ( H ) . Besides random graphs, explicit constructions of η -uniform graphs of constant average degreeare also known. One class of examples of such graphs are ( n, d, λ )-graphs, for suitably chosenparameters. Indeed, the well known Expander Mixing Lemma [4] states that for every ( n, d, λ )-graph G and for every U, V ⊆ V ( G ) it holds: (cid:12)(cid:12)(cid:12)(cid:12) e G ( U, V ) − d | U || V | n (cid:12)(cid:12)(cid:12)(cid:12) ≤ λ q | U || V | . (1)From this, one can see that every ( n, d, λ )-graph is η -uniform for η = 2 q λd . Hence, for a fixed d , the parameter λ is accountable for the uniformity of the distribution of the edges of a d -regular graph. But how small can λ be in terms of d , so that there exists a ( n, d, λ )-graph?One can show that λ = Ω( √ d ) for every such graph whenever d < . n , and there are knownconstructions of d -regular graphs for which λ attains this bound, and n is arbitrarily large. Thisprovides us examples of bounded degree graphs, which are η -uniform (for η ∼ d − / ). For severalconstructions of such graphs, see, e.g., [42]. As discussed above, there are known constructionsof such graphs which have logarithmic girth, showing that our result is asymptotically tightwith respect to the bound on σ . Indeed, if H is a triangle, and G is a graph with girth strictlylarger than c log n , then G does not contain H σ for any σ bounded from above by c log n/ universality result, meaning that the η -uniform graph inquestion is k -Ramsey for all graphs in the class we are interested in, hence our theorem confirmsPak’s conjecture in a strong way. Furthermore, from our proof it can be seen that we actuallyfind a monochromatic subgraph of the graph we color, which contains all described subdividedgraphs. Extending the definition in [39], we say that a graph G is k -partition universal fora class of graphs F if for every k -coloring of the edges of G , there exists a monochromaticsubgraph of G which contains a copy of every graph in F . Under this framework, we actuallyprove that the graph we color is up to a constant factor the optimal k -partition universal graphfor the class of all described subdivisions of graphs. For further universality-type results inRamsey theory see for example [18, 22, 38, 39]. For a given graph G and a collection of k disjoint pairs of vertices ( a i , b i ) from G , can we findfor each i a path from a i to b i , such that the found paths are all vertex-disjoint? This decisionproblem is N P -complete [30] when G is allowed to be an arbitrary graph. Furthermore, itremains N P -complete, even when G is restricted to be in the class of planar graphs. Forfixed k , it is shown to be in P [52]. A variant of this problem in random graphs was studiedindepedently by Hochbaum [33], and by Shamir and Upfal [53]. Both papers proved that for afixed set of at most O ( √ n ) disjoint pairs of vertices in the random graph G ( n, m ), with highprobability (whp) there exist vertex-disjoint paths between every pair if m > Cn log n , for aconstant C >
1. Subsequently, Broder, Frieze, Suen and Upfal [13] improved this result:
Theorem ([13]) . There exist α, β > , such that whp the following holds. Let G = G ( n, m ) for m = (log n + ω (1)) n , and let d = 2 m/n . For every collection F of at most α n log d log n disjoint pairsof vertices ( a i , b i ) in G , there exists a path for every i connecting a i to b i , such that all pathsare vertex-disjoint, if the following condition is satisfied: | N G ( v ) ∩ ( A ∪ B ) | < βd G ( v ) , where A = ∪ i { a i } and B = ∪ i { b i } . G ( n, m ) are at distance Ω(log n/ log d ),so in general one can hope to connect at most O ( n log d log n ) pairs ( a i , b i ) in a graph on n vertices.Furthermore, the pairs ( a i , b i ) are not fixed before generating G ( n, m ), but are rather chosenadversarily after having exposed a random graph G ∼ G ( n, m ). The last constraint is alsooptimal up to a constant factor — if the adversary chooses a and b to be at distance 2, andthen chooses all neighbours of a to be in other pairs from F , then obviously one cannot find therequested disjoint paths. The bound on the number of edges m is also asymptotically optimal,as this many edges are needed for G to be connected whp.Changing the focus to the sparse(r) setting, Alon and Capalbo [3] studied graphs with constantaverage degree with good expansion properties. In particular, they proved that for any graph G which is a d -blowup of a ( n, d, λ )-graph with a small spectral ratio, and any collection of O (cid:16) nd log d log n (cid:17) pairs of vertices in G which satisfy a similar local condition like in [13], one canconnect those pairs with vertex-disjoint paths. The number of pairs is optimal up to a constantfactor, and they provide a polynomial time algorithm for finding them.The argument of Alon and Capalbo does not allow to control the length of the paths found bythe algorithm. Accordingly, they ask for a similar result where the length of the paths betweeneach pair is at most O (log n ). In Section 3.2, we prove such a result (Theorem 1.3). Furthermore,we do it not only for blowups, but directly for ( n, d, λ )-graphs for λ < d . . We get the optimaldependency on n and d , both for the number of pairs and for the upper bound on the length ofthe paths. Theorem 1.3.
Let ε > , and let G be an ( N, D, λ ) -graph, with λ < D − ε / and D ε > .Let P = { a i , b i } be a collection of at most εN log D
160 log N disjoint pairs of vertices in G , such that | N G ( x ) ∩ ( A ∪ B ) | ≤ D for every x ∈ V ( G ) , where A = ∪ i { a i } , B = ∪ i { b i } . There existsa polynomial time algorithm to find vertex-disjoint paths in G between every pair of vertices { a i , b i } , such that the paths are of equal length which is less than log Nε log D . These results are closely related to the study of non-blocking networks, which arise in a varietyof applications, including construction of communication networks and distributed-memory ar-chitectures. For some results see, e.g., [26, 27, 49]. In contrast to our results, the graphs whichare usually considered here have pre-determined sets of vertices ("inputs" and "outputs") fromwhich the pairs are chosen, while the pairs in our result can be chosen by an adversary, butin such a way that they satisfy an essentially minimal local property. Besides that, the pathlengths in some constructions of non-blocking networks are also of optimal O (log n ) size [6].Hence, in some sense our results are a common generalization of [6] and [3], as we both allowthe adversary to choose the pairs, and our paths are logarithmic in size, although our algorithmis less efficient than the one in [6], and is not online in the same sense like in [3].A lot of attention has also been paid to the edge-disjoint paths problem. For a short survey, see[29], and for a more recent result on edge-disjoint paths in sufficiently strong expander graphssee [3]. In Section 2 we show two versions of our main embedding technique — in Section 2.1 we show thenon-algorithmic version of it, and in Section 2.2 we give an algorithmic version of the technique.In Section 3, we prove our main results — Theorem 1.2 (the resolution of Pak’s conjecture)in Section 3.1, and Theorem 1.3 (vertex-disjoint paths in (
N, D, λ )-graphs) in Section 3.2. Insection 4 we give some concluding remarks. 5 otation.
We follow standard graph theoretic notation. In particular, given a graph G and a vertex x ∈ V ( G ), we denote by N G ( x ) the neighborhood of x in G . Similarly, for asubset of vertices X ⊆ V ( G ) we denote by N G ( X ) the external neighborhood of X , that is N G ( X ) = ( S x ∈ X N G ( x )) \ X . By ∂ G ( x ) we denote the set of edges incident with vertex x in G . Given disjoint subsets of vertices A, B ⊆ V ( G ), we denote by e G ( A, B ) the numberof edges with one endpoint in A and the other in B , and with d G ( A, B ) = e G ( A, B ) / | A || B | the density of such a induced bipartite graph. We denote by v ( G ) the number of vertices of G , and by e ( G ) the number of edges of G . Given graphs G and H , we say that a mapping φ : V ( H ) → V ( G ) is an embedding , with the notation φ : H ֒ → G , if it is injective and preservesedges of H (i.e. if { v, w } ∈ E ( H ) then { φ ( v ) , φ ( w ) } ∈ E ( G )). For an embedding φ : H ֒ → G and subsets S ⊆ V ( H ) , S ⊆ V ( G ) we denote by φ ( S ) the image of S under φ , and by φ − ( S ) the preimage of S under φ , i.e. φ ( S ) = { y ∈ V ( G ) | ∃ x ∈ S : φ ( x ) = y } , and φ − ( S ) = { x ∈ V ( H ) | φ ( x ) ∈ S } . We omit floors and ceilings whenever it is not crucial.Given two constant ε and α , we use somewhat informal notation ε ≪ α to denote that ε issufficiently small compared to α . We denote by log n the natural logarithm of n . Now we describe the main embedding machinery behind our proofs. It relies on the idea ofFriedman and Pippenger, used for embedding trees in expanders vertex by vertex, by maintain-ing a certain invariant. An algorithmic version of this technique was presented by Dellamonicaand Kohayakawa, based on a result about an online matching game by Aggarwal et al. [1].In the following two subsections, we give two Friedman-Pippenger type embedding theorems,non-algorithmic and algorithmic, enhanced with a roll-back idea, which allows us to sequentiallyretrace previously performed embedding steps. While the algorithmic result requires the hostgraph to have stronger expansion properties, it also enables us to embed larger graphs thanwith the technique described in Section 2.1.
We start with a standard definition of expansion.
Definition 2.1.
Let s ∈ N and K >
0. We say that a graph G is ( s, K ) -expanding if for everysubset X ⊆ V ( G ) of size | X | ≤ s we have | N G ( X ) | ≥ K | X | .In order to develop our machinery, we define the notion of an ( s, D ) -good embedding. Definition 2.2.
Let G be a graph and let s, D ∈ N . Given a graph F with maximum degreeat most D , we say that an embedding φ : F ֒ → G is ( s, D ) -good if | N G ( X ) \ φ ( F ) | ≥ X v ∈ X h D − deg F ( φ − ( v )) i (2)for every X ⊆ V ( G ) of size | X | ≤ s . Here we slightly abuse the notation by setting deg F ( ∅ ) := 0,i.e. if a vertex v ∈ V ( G ) is not used by φ to embed F , then we set deg F ( φ − ( v )) = 0.We remark that the notion of a good embedding is the same as the one used by Friedman andPippenger [28]. The following is implicit in [28]. Theorem 2.3.
Let F be a graph with ∆( F ) ≤ D and v ( F ) < s , for some D, s ∈ N . Supposewe are given a (2 s, D +1) -expanding graph G and a (2 s, D ) -good embedding φ : F ֒ → G . Thenfor every graph F ′ with v ( F ′ ) ≤ s and ∆( F ′ ) ≤ D which can be obtained from F by successivelyadding a new vertex of degree , there exists a (2 s, D ) -good embedding φ ′ : F ′ ֒ → G which extends φ . m, D )-goodness. Whileeasy to prove, this observation [36] turns out to yield a powerful method for connecting verticesin expanding graphs. It has also been utilized in the recent paper by Montgomery [47] forembedding spanning trees in random graphs. Lemma 2.4.
Suppose we are given graphs G and F and an ( s, D ) -good embedding φ : F ֒ → G ,for some s, D ∈ N . Then for every graph F ′ obtained from F by successively removing a vertexof degree , the restriction φ ′ of φ to F ′ is also ( s, D ) -good.Proof. We show that the statement holds for the case where F ′ is obtained from F by removinga single vertex v ∈ V ( F ) of degree 1. The lemma then follows by iterating it.Let φ ′ be a restriction of F to such F ′ , and let w ∈ F ′ denote the unique neighbor of v . If X ⊆ V ( G ) does not contain φ ( w ) then the right hand side of (2) does not change. Otherwise(if φ ( w ) ∈ X ) the right hand side of (2) increases by 1 (as the degree of w in F ′ is one less thanit was in F ). However, as φ ( v ) is no longer occupied (i.e. φ ( v ) / ∈ φ ′ ( F ′ )) and φ ( v ) ∈ N G ( φ ( w )),the left hand side also increases by one, hence the inequality again holds. In this section we prove an algorithmic version of the embedding technique provided by Theorem2.3 and Lemma 2.4 from Section 2.1. We start with a description of an online matching game,to which we reduce our embedding problem.Let m ≥ H = ( U ∪ V, E ). In thebeginning we set M ( the current matching ) to be empty. At each step an adversary chooses avertex x ∈ U which is not covered by M , and we match it to some free vertex in V to extend M . After each step the adversary is allowed to remove any number of edges from the currentmatching M , but at most m times in total during the game. In [1, Lemma 2.2.7], Aggarwal et al.describe a polynomial time algorithm which finds a matching of size n , against any adversary,if H satisfies the property that for each X ⊂ U of size | X | ≤ n , even if we remove at most halfof the edges incident to every vertex in X , there are still at least 2 | X | neighbors of X in theobtained graph. Theorem 2.5 ([1], Aggarwal et al.) . Let H = ( U ∪ V, E ) be a bipartite graph and let n, m ∈ N ,such that for every X ⊆ U of size | X | ≤ n and for every F ⊆ E such that | F ∩ ∂ H ( x ) | ≤ d H ( x ) / for every x ∈ X , we have that | N H − F ( X ) | ≥ | X | . Then there is an algorithm which finds amatching of size n against any adversary, if the adversary is allowed to remove edges from thematching at most m times in total during the game. Furthermore, the number of operationswhich the algorithm performs is polynomial in m + | V ( H ) | . Definition.
We say that a graph G = ( V, E ) has property P α ( n, d ) if for every X ⊆ V ofsize | X | ≤ n and every F ⊆ E such that | F ∩ ∂ G ( x ) | ≤ α · d G ( x ) for every x ∈ X , we have | N G − F ( X ) | ≥ d | X | . Definition 2.6.
Given a graph G , a subset of vertices P ⊆ V ( G ), and natural numbers n, m, d ∈ N , we define the following online game, which we call the ( G, P, n, m, d ) -forest building game .At each step there is a forest T ⊆ G (initially T := ( P, ∅ )) with less than n edges in G , and theadversary requests a vertex v ∈ T such that d T ( v ) < d and we are supposed to find a neighbor of v in V ( G ) − V ( T ), hence extending T by a new leaf. The adversary is is allowed to successivelyremove any number of vertices of degree 1 in T after every step, but he is allowed to do so atmost m times in total, and none of the removed vertices are allowed to be in P . We win if atsome point T has n edges.The next theorem gives a handy tool for embedding forests algorithmically in a robust way.In comparison to the technique presented in Section 2.1, here we require a stronger notion of7xpansion (the P α ( n, d )-property) for the host graph, but the graphs we are embedding canhave more vertices than before. The idea of the proof is similar to the one in [20]. Theorem 2.7.
Let α, β > with α − β ≥ / and let G be a graph with property P α ( n, d ) . Let P be a non-empty subset of vertices P ⊆ V ( G ) , such that for every vertex x ∈ V ( G ) it holdsthat | N G ( x ) ∩ P | ≤ β · d G ( x ) . Then there is an algorithm which wins the ( G, P, dn, m, d ) -forestbuilding game after performing a number of operations polynomial in m + | V ( G ) | .Proof. In order to use Theorem 2.5, we construct the following auxiliary graph. Let H be abipartite graph with classes U = V ( G ) × [ d ] and V = { ¯ v | v ∈ V ( G ) − P } . In other words, U consists of d copies of V ( G ) , and V is a copy of V ( G ) − P . Two vertices ( u, j ) ∈ U and ¯ v ∈ V are adjacent iff { u, v } is an edge in G . Now we show that H satisfies the condition of Theorem2.5 (with dn instead of n ).Let X ⊆ U be of size | X | ≤ dn , and F ⊆ E ( H ) be such that | F ∩ ∂ H ( x ) | ≤ d H ( x ) / for every x ∈ X . We want to show that | N H − F ( X ) | ≥ | X | . By the pigeonhole principle, one of the d copies of V ( G ) in U contains at least | X | /d elements from X , or in other words, there is an i ∈ [ d ] such that the set X i := { ( u, i ) | ( u, i ) ∈ X } is of size | X i | ≥ | X | /d . Let Y be an arbitrarysubset of X i of size exactly ⌈| X | /d ⌉ , and let Y ′ = { u | ( u, i ) ∈ Y } ⊆ V ( G ) .We also define F ′ ⊆ E ( G ) as follows: F ′ = n { u, v } ∈ E ( G ) | u ∈ Y ′ , v / ∈ Y ′ , and { ( u, i ) , ¯ v } ∈ F o . Let G ′ be the graph obtained from G by deleting all edges in F ′ and by deleting all edges whichhave one vertex in Y ′ and the other in P \ Y ′ . Note the following facts:( i ) | N H − F ( Y ) | ≥ | N G ′ ( Y ′ ) | ,( ii ) d G ′ ( x ) ≥ (1 − α ) d G ( x ) for all x ∈ Y ′ .The first claim is true as for every vertex v ∈ N G ′ ( Y ′ ) there is a vertex ( u, i ) ∈ Y such that { ( u, i ) , ¯ v } is an edge in H − F . For the second claim, notice that after deleting F ′ from G eachvertex x ∈ Y ′ loses at most half of its edges, and by deleting the edges incident to P \ Y ′ , it losesat most another β · d G ( x ) edges, which is in total at most (1 / β ) d G ( x ) ≤ α · d G ( x ) edges.It follows from the second claim and from | Y ′ | = ⌈| X | /d ⌉ ≤ ⌈ nd/d ⌉ = n (and from the assump-tion that G has the P α ( n, d ) -property), that | N G ′ ( Y ′ ) | ≥ d | Y ′ | ≥ | X | . Together with ( i ) thisimplies | N H − F ( X ) | ≥ | N H − F ( Y ) | ≥ | X | .Now we reduce our forest building game on the graph G to the matching game on the graph H . At the beginning our initial forest is set to be the empty graph on P , i.e. T := ( P, ∅ ) ⊆ G .We also set our auxiliary matching M in H to be empty in the beginning. During the game M and T will have the same number of edges. In each step the adversary requests a vertex u ∈ T such that d T ( u ) < d , and we want to find a vertex v in N G ( u ) \ V ( T ) which extends T , suchthat { u, v } is a new leaf in T . In order to do this, we find a vertex ( u, j ) in H for some j ≤ d ,which is not covered by M (in the next paragraph we show that such a vertex exists), and weextend M by finding a match ¯ v ∈ V for ( u, j ) , using the algorithm from Theorem 2.5. Now weadd the edge ( u, v ) (which is in G by the definition of H ) to T . Note also that v was not in T before, as v certainly is not in P (by the definition of ¯ v ), and for every other vertex x ∈ T , thevertex ¯ x is covered by M , as x has been added to T by the same procedure, so ¯ x = ¯ v .When the adversary wants to delete an edge ( u, v ) (where v is of degree in T ) from T , thenwe also delete the corresponding edge { ( u, j ) , ¯ v } from M . Note that if at any step the adversaryrequests a vertex u such that d T ( u ) < d , then a vertex of the form ( u, i ) (for some i ∈ [ d ] ) hasbeen used only at most d − times by the current matching M , so it is valid to assume that8n each step we can find such a vertex which is not covered by M . Since the algorithm findsa forest T with dn edges at the same point when M contains dn edges, and we remove edgesfrom the matching only at most m times in total, thanks to Theorem 2.5, we are done. Before we start with the proof of Theorem 1.2, we state a few preliminary results which willhelp us find a subgraph with good expansion properties in the edge colored graph in question.
The following lemma tells us that if in a graph all sets of a specified size expand well, we candelete relatively few vertices, so that in the remaining graph all smaller sets also expand well.For related results see for example [40]. A similar statement also appeared in [45].
Lemma 3.1.
Let G be a graph such that | N G ( X ) | ≥ Ks for every subset X ⊆ V ( G ) of size | X | = s , for some s, K ∈ N . Then there exists a subset B ⊆ V ( G ) of size | B | < s such that G − B is ( s, K ) -expanding.Proof. Let B ⊂ V ( G ) be a largest set such that | N G ( B ) | < K | B | and | B | < s (or B = ∅ ifno such set exists). We show that H = G − B is ( s, K ) expanding. Let X ⊆ V ( H ) be anarbitrary non-empty set of size | X | ≤ s and suppose | N H ( X ) | < K | X | . Then | N G ( X ∪ B ) | The proof of Theorem 1.2 combines results from Section 2.1 with a sparse version of Szemerédi’sregularity lemma for multicolored graphs (or rather its corollary given shortly). Definition. Given a graph G and disjoint subsets U, W ⊆ V ( G ) , we say that the pair ( U, W ) is ( G, ε, p ) - regular for some ε, p ∈ (0 , if | d G ( U ′ , W ′ ) − d G ( U, W ) | ≤ εp for every U ′ ⊆ U of size | U ′ | ≥ ε | U | , and W ′ ⊆ W of size | W ′ | ≥ ε | W | . Remark 3.2. If U ′ ⊆ U and W ′ ⊆ W are as above and d G ( U, W ) > εp , then there existsat least one edge between U ′ and W ′ in G , as otherwise d G ( U ′ , W ′ ) = 0 , which contradicts | d G ( U ′ , W ′ ) − d G ( U, W ) | ≤ εp . It follows that | N G ( U ′ ) | > (1 − ε ) | W | . The following corollary of Szemerédi’s regularity lemma was proven in [32, Lemma 3.4]. Lemma 3.3. For every k ≥ and < ε < , there exist µ, η > such that the following holds:Suppose G = ( V, E ) is an η -uniform graph with n vertices and density p = e ( G ) / (cid:0) n (cid:1) > , andlet E = E ·∪ E ·∪ . . . ·∪ E k be an k -edge-coloring of G . Then, for some ≤ z ≤ k , there existpairwise disjoint subsets V , V , V ⊆ V of size | V i | = µn such that(a) ( V i , V j ) is ( G z , ε, p ) -regular, where G z = ( V, E z ) , and(b) d G z ( V i , V j ) ≥ p | V i || V j | / k , or every ≤ i < j ≤ . We are ready to prove Theorem 1.2, which we restate here. Theorem 1.2. For every k, D ∈ N and for every δ > , there exist η, α, C > , such thatthe following holds for every η -uniform graph G with n vertices and m ≥ Cn edges: every k -edge-coloring of G contains a monochromatic copy of every graph H σ , where H is a graph withmaximum degree at most D , v ( H σ ) ≤ αn and σ ( e ) ≥ δ log n for every e ∈ E ( H ) . Proof of Theorem 1.2. Let µ = µ ( k, ε ) and η = η ( k, ε ) > be given by Lemma 3.3 for asufficiently small constant ε ≪ D − , k − . Also assume w.l.o.g. that D ≫ /δ . Suppose we aregiven an η -uniform graph G with n vertices and a k -edge-coloring E ( G ) = E ·∪ E ·∪ . . . ·∪ E k ,and let ≤ z ≤ k and V , V , V ⊆ V ( G ) be obtained by applying Lemma 3.3. In the rest ofthe proof we show that Γ = ( V ( G ) , E z ) contains H σ for every H satisfying conditions of thetheorem with α = εµ . Prepare Γ . Let t = | V i | = µn . Let Γ ′ = Γ[ V , V ] be a bipartite subgraph of Γ induced by V and V . From (Γ , ε, p ) -regularity of ( V , V ) and from the assumption ε ≪ /k, /D , weconclude (Remark 3.2) that for every subset X ⊆ V (Γ ′ ) of size | X | = 2 s , where s = 2 D εt, we have | N Γ ′ ( X ) | ≥ t − εt − | X | ≥ t/ ≥ D + 3) | X | . Therefore, by Lemma 3.1 there exists a subset B ⊆ V (Γ ′ ) of size | B | = s such that Γ B = Γ ′ \ B is (2 s, D + 3) -expanding. Let V ′ = V \ B and V ′ = V \ B , so that Γ B = Γ B [ V ′ , V ′ ] . Most of H σ will be embedded using Γ B and the machinery from Section 2.1, with occasional help fromset V . Embed H . Consider a graph H with maximum degree D and let σ : E ( H ) → N be a func-tion such that v ( H σ ) = v ( H ) + P e ∈ E ( H ) σ ( e ) < εt and σ ( e ) ≥ δ log n for every e ∈ E ( H ) .Let ( e , . . . , e m ) be an arbitrary ordering of the edges of H , and for each ≤ i ≤ m set H i = ( V ( H ) , { e , . . . , e i } ) . Note that H is just an empty graph on the vertex set V ( H ) . Weinductively show that for each ≤ i ≤ m there exists an embedding φ i : H σi ֒ → Γ such that thefollowing holds:(1) φ i ( V ( H )) ⊆ V ′ , and(2) the restriction of φ i to F i = φ − i ( V (Γ B )) , denoted by f i : F i ֒ → Γ B , is (2 s, D ) -good.Let us first prove the base case i = 0 . Note that H σ = H is an empty graph on the vertexset V ( H ) . Let a be a vertex (some new auxiliary vertex not used before) and v ∈ V , andset φ ′ ( a ) = v . As Γ B is (2 s, D + 3) -expanding, it is easy to see that φ ′ is a (2 s, D + 2) -goodembedding of a graph consisting of a single vertex. Let us extend such a one-vertex graph to apath P of length εt . By Theorem 2.3, there exists an (2 s, D + 2) -good embedding φ ′ : P ֒ → Γ B .Consider an arbitrary bijection between V ( H ) and the set of odd vertices in P (i.e. the firstvertex, third vertex, etc.). As φ ′ ( a ) is mapped into V ′ , all these vertices are also necessarilymapped into V ′ . Together with φ ′ , such a bijection gives an embedding φ : H ֒ → Γ B with φ ( V ( H )) ⊆ V ′ . As φ ′ was a (2 s, D + 2) -good embedding, it is easy to verify that φ is a (2 s, D ) -good embedding. 10uppose the induction holds for some i < m and let e i +1 = { a, b } . In short, we need to finda path from φ i ( a ) to φ i ( b ) of length σ ( e i +1 ) , such that the part of it that goes through Γ B maintains (2 s, D ) -goodness. In the proof we use auxiliary parameters ℓ , ℓ , h ∈ N , definedas follows: choose h ∈ N to be the smallest integer such that ( D − h ≥ εt , and set ℓ = ⌊ σ ( e i +1 ) / ⌋ − h − and ℓ = ⌈ σ ( e i +1 ) / ⌉ − h − . Note that ℓ , ℓ > since ⌊ σ ( e i +1 ) / ⌋ ≥⌊ δ log n/ ⌋ ≥ log D − n > h + 2 , where we used /ε ≫ D ≫ /δ .Let F i = φ − i (Γ B ) be the part of H σi embedded into Γ B , and f i be the restriction of φ i to F i .We start by constructing the graph F ′ i in two steps: First attach to F i two paths of lengths ℓ and ℓ , one rooted in a and the other in b , and let a ′ and b ′ denote other ends of suchpaths. Then attach two complete ( D − -ary trees of depth h , one rooted in a ′ and the otherin b ′ . Let us denote the set of leaves of these trees by L a and L b , respectively, and note that | L a | = | L b | = ( D − h ≥ εt by the choice of h . Such trees have less than ( D − h +1 ≤ ( D − εt vertices each, which together with a trivial bound ℓ ≤ ℓ < σ ( e i +1 ) < v ( H σ ) implies v ( F ′ i ) ≤ v ( F i ) + 2( ℓ − 1) + 2 · (( D − εt − ≤ v ( H σ ) + 2 v ( H σ ) + 2( D − εt < s. Assuming D ≥ each vertex has degree at most D in F ′ i and, by its definition, F ′ i can beconstructed from F i by successively adding a vertex of degree . Therefore, we can applyTheorem 2.3 to obtain a (2 s, D ) -good embedding f ′ i : F ′ i ֒ → Γ B which extends f i .Every vertex in L a is at distance exactly ℓ + h from a , and every vertex in L b is at distanceexactly ℓ + h from b . Thus f ′ i ( L a ) ⊆ V ′ j and f ′ i ( L b ) ⊆ V ′ j for some j , j ∈ { , } . Next,we find a path of length from f ′ i ( L a ) to f ′ i ( L b ) with the internal vertex lying in V . From (Γ , ε, p ) -regularity of the pairs ( V , V ) and ( V , V ) , and | f ′ i ( L a ) | , | f ′ i ( L b ) | ≥ εt , we know thatall but at most εt vertices in V \ φ i ( H σi ) are adjacent to both f ′ i ( L a ) and f ′ i ( L b ) . As | V | = t and v ( H σ ) < εt , this implies that there exists a free vertex in V adjacent both to f ′ i ( L a ) and f ′ i ( L b ) , which gives a desired path of length .To summarize, we have found a path P ( x, y ) of length from f ′ i ( x ) to f ′ i ( y ) , for some x ∈ L a and y ∈ L b , with the internal vertex avoiding V ∪ V and φ i ( H σi ) . By Lemma 2.4, the restrictionof f ′ i to the graph obtained by removing all newly added vertices to F i which do not lie eitheron the path from x to a or from y to b is (2 s, D ) -good. Together with the path P ( x, y ) , thisdefines an embedding φ i +1 of H σi +1 into Γ . Theorem 2.7 provides a framework for embedding forests (in polynomial time) into graphs withcertain expansion properties, while allowing arbitrary leaf deletions along the way. We presentan application of this result to the classical problem of finding vertex-disjoint paths betweengiven pairs of vertices in graphs.Now we state the key result of this subsection. Theorem 1.3 (stated in the introduction) willthen follow directly from the properties of ( N, D, λ ) -graphs. Theorem 3.4. Let G be a graph with the P α ( n, d ) property for ≤ d < n , and such that forevery two disjoint U, V ⊆ V ( G ) of sizes | U | , | V | ≥ n/ d there exists an edge between U and V . Let P = { a i , b i } be a collection of at most dn log d n disjoint pairs of vertices in G , such that | N G ( x ) ∩ ( A ∪ B ) | ≤ βd G ( x ) for every x ∈ V ( G ) , where A = ∪ i { a i } , B = ∪ i { b i } . If α − β ≥ / then there exists a polynomial time algorithm to find vertex-disjoint paths in G between everypair of vertices { a i , b i } , such that the length of each path is ⌈ log( n/ d )log( d − ⌉ + 1 .Proof. By Theorem 2.7, there is an algorithm which works in time polynomial in V ( G ) , andwins the ( G, P, nd, n , d ) -forest building game. We construct the required disjoint paths one byone as follows. Let h be the smallest integer such that ( d − h > n d .11or the first pair { a , b } we find two disjoint complete ( d − -ary trees of depth h in G , rootedat a and b , using the algorithm for winning the forest building game. Since both trees havemore than n d leaves, there is an edge connecting these sets of leaves, thus creating a path(between a and b ) of length h + 1 . Remove from our current forest all other edges which donot lie on this path. We continue in the same fashion, by finding two complete ( d − -ary treesrooted at a and b (disjoint from the path connecting a and b ), then finding a connectingedge between the sets of leaves, and removing all edges from the ( d − -ary trees, which do notlie on the found path. We delete the edges successively, by always removing the edges whichare incident with vertices of degree , just like in the forest building game.We do this procedure for every pair of vertices, and note that we can do this as at any givenpoint the current forest which we use for our argument has at most dn log d n · (2 h + 1) + 2 · · n < dn n < dn edges, where the first term is a bound on the total number of edges used in previous paths, andthe second one bounds the number of edges in the current ( d − -ary trees we use. Furthermorewe delete vertices of degree 1 at most | A ∪ B | · n < n times. This completes the proof.The following result can be derived from the Expander Mixing Lemma through rather routinecalculations. Lemma 3.5 ([20], Lemma 2.7) . Let G be an ( N, D, λ ) -graph and let d, n be positive integers. G has property P α ( n, d ) for α > if the following holds: − α > n (1 + 4 d )2 N + λD (1 + √ d ) . We are ready to give the promised proof of Theorem 1.3. Theorem 1.3. Let ε > , and let G be an ( N, D, λ ) -graph, with λ < D − ε / and D ε > .Let P = { a i , b i } be a collection of at most εN log D 160 log N disjoint pairs of vertices in G , such that | N G ( x ) ∩ ( A ∪ B ) | ≤ D for every x ∈ V ( G ) , where A = ∪ i { a i } , B = ∪ i { b i } . There existsa polynomial time algorithm to find vertex-disjoint paths in G between every pair of vertices { a i , b i } , such that the paths are of equal length which is less than log Nε log D .Proof. Let n = N/ d and d = D ε/ . From Lemma 3.5 we see that G has the P / ( n, d ) -property.Furthermore, by the Expander Mixing Lemma (eq. (1)), we have that for sets U, V ⊆ V ( G ) ofsize at least n d it holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) e G ( U, V ) − Dn N d (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ λ n d . which, together with λ < D − ε / gives e G ( U, V ) > . Applying Theorem 3.4 to G completesthe proof; here are the final calculations. • Number of pairs: dn log d n = N log D ε/ · N/ d ) = εN log D 160 log( N/ d ) > εN log D 160 log N ; • Length of paths: (cid:24) log( n/ d )log( d − (cid:25) + 1 = 2 & log( N/ d )log( D ε/ − ' + 1 ≤ Nε log D . Concluding remarks We presented two embedding techniques (algorithmic and non-algorithmic) for embedding graphsof large girth into sparse expanders. Both are based on the embedding result by Friedman andPippenger, enhanced with a roll-back idea which allows retracing previous embedding steps. Weshowed two applications of these techniques: • We proved that the size-Ramsey number of logarithmic subdivisions of bounded degreegraphs is linear in their order; • For a given ( n, d, λ ) -graph with relatively small spectral ratio and any collection of c n log d log n disjoint pairs of vertices which satisfy a natural local condition, we gave a polynomial timealgorithm which finds vertex disjoint paths of (the same) logarithmic length between eachpair.The first result answers a question of Pak [48], and the second one answers an offline versionof a question of Alon and Capalbo [3]. With regards to the latter result, our offline algorithmcan be made online in the following sense: instead of all the pairs being given in advance, theadversary can choose a set S of vertices of size | S | = c n log d log n (which satisfies the same localcondition as before, and is given after G is exposed). 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