Low-distortion embeddings of graphs with large girth
aa r X i v : . [ m a t h . M G ] A ug Low-distortion embeddings of graphs with large girth
Mikhail I. Ostrovskii ∗ October 28, 2018
Abstract.
The main purpose of the paper is to construct a sequence of graphs of constant degreewith indefinitely growing girths admitting embeddings into ℓ with uniformly bounded distortions.This result solves the problem posed by N. Linial, A. Magen, and A. Naor (2002). Primary: 46B85; Secondary: 05C12, 54E35
Definition 1.
Let
C < ∞ . A map f : ( X, d X ) → ( Y, d Y ) between two metric spacesis called C - Lipschitz if ∀ u, v ∈ X d Y ( f ( u ) , f ( v )) ≤ Cd X ( u, v ) . A map f is called Lipschitz if it is C -Lipschitz for some C < ∞ . For a Lipschitzmap f we define its Lipschitz constant byLip f := sup d X ( u,v ) =0 d Y ( f ( u ) , f ( v )) d X ( u, v ) . A map f : X → Y is called a C -bilipschitz embedding if there exists r > ∀ u, v ∈ X rd X ( u, v ) ≤ d Y ( f ( u ) , f ( v )) ≤ rCd X ( u, v ) . (1)A bilipschitz embedding is an embedding which is C -bilipschitz for some C < ∞ .The smallest constant C for which there exist r > distortion of f . (It is easy to see that such smallest constant exists.)The infimum of distortions of all embeddings of a finite metric space X into theBanach space ℓ is called the ℓ distortion of X and is denoted c ( X ).The ℓ distortion of finite metric spaces plays an important role in the theory ofapproximation algorithms, see [Lin02], [LLR95], [Mat02], [Mat05], and [Nao10].Our main purpose is to solve the following problem suggested in [LMN02, p. 393]and repeated in [Lin02, Open Problem 7] and [Mat10, Problem 2.3]: Does thereexist a sequence of k -regular graphs, k ≥
3, with indefinitely growing girths anduniformly bounded ℓ distortions? (All graphs mentioned in this paper are endowedwith their shortest path distance.) We are going to show that such sequences exist.The construction of this paper is inspired by the paper [AGS11+]. Recall thatthe girth g ( G ) of a graph G is the length of a shortest cycle in G . We start with ∗ Participant, NSF supported Workshop in Analysis and Probability, Texas A & M University sequence of k -regular graphs { G n } with indefinitely increasing girths g ( G n ), suchthat g ( G n ) ≥ c diam( G n ) (2)for some absolute constant c . Existence of such sequences of graphs is known forlong time, see [Bol78, Chapter III, § G in the sequence { G n } we consider its lift e G in the sense of the papers[AL06] and [DL06]. (We would like to warn the reader that somewhat different ter-minology (graph covers, voltage graphs) is used in other publications on the topic,such as [AGS11+], [GT77], and [GT87].) The particular version of the lift which weuse is the same as the lift used in [AGS11+], but it is applied to a different sequenceof graphs. Also we need somewhat stronger estimates than those which were suffi-cient for [AGS11+]. Another difference of our presentation from the presentation in[AGS11+] is that we try to keep the presentation as elementary as possible, withoutassuming any topological and group-theoretical background of the reader. We useonly some basic notions of graph theory and the definition of the space ℓ . We hopethat our graph-theoretical terminology is standard, readers can find all unexplainedterminology in [BM08]. Definition 2.
Let L be a finite set. A lift e G of a graph G = ( V ( G ) , E ( G )) is agraph with vertex set V ( e G ) = V ( G ) × L . The edge set e G is the union of perfectmatchings corresponding to edges of E ( G ). The matching corresponding to an edge uv matches { u } × L with { v } × L .Definition 2 immediately implies that there are well-defined projections E ( e G ) → E ( G ) and V ( e G ) → V ( G ): edges of the matching corresponding to uv are projectedonto uv and vertices of { u } × L are projected onto u . We denote both of theprojections by π . It is clear from the definition that the degrees of all vertices in e G whose projection in G is u are the same as the degree of u . In particular, any lift ofa k -regular graph is k -regular. Remark . It is easy to see that for each walk { e i } ni =1 in G and each vertex e u ∈ V ( e G )of the form e u = ( u, ℓ ) with ℓ ∈ L and u being the initial vertex of the walk { e i } ni =1 ;there is a uniquely determined lifted walk { e e i } ni =1 in e G for which π ( e e i ) = e i and e u isthe initial vertex. Remark . It is clear that if a walk in G has an edge e which is backtracked (thatis, the walk contains two consecutive edges e ), then the corresponding edge in thelifted walk is also backtracked.Remark 4 implies that the projection of a cycle in e G to G cannot be such thatits edges induce in G a subgraph having vertices of degree 1. In particular, thegraph induced by edges of the projection of a cycle in e G contains cycles in G . Thisimmediately implies g ( e G ) ≥ g ( G ). e apply the lift construction to the graphs { G n } mentioned above. The factthat we get k -regular graphs with indefinitely increasing girths follows immediatelyfrom the observations which we just made. It remains to specify lifts for which thereare suitable estimates for ℓ distortions of the obtained graphs. The bounds for thedistortions which we get are in terms of the constant c in (2).For each G ∈ { G n } ∞ n =1 we do the following. We pick a spanning tree T in G andlet S be the set of edges of G which are not in T . We let L to be the set { , } S ,so each element of L can be regarded as a { , } -valued function on S . For each uv ∈ E ( G ) we need to specify a perfect matching of u × L and v × L . To specifythe perfect matching it suffices, for each edge in E ( G ), to pick a bijection of the set L . We do this in the following way: • If e ∈ E ( T ) (that is, if edge is in the spanning tree which we selected), thenthe corresponding bijection is the identical mapping on L . • If e ∈ S , then the bijection maps each function f on S to the function h , whichhas the same values as f everywhere except the edge e , and on the edge e itsvalue is the other one (recall that we consider { , } -valued functions).We denote the graphs obtained from { G n } using such lifts by { e G n } . The followingtheorem is the main result of this paper. Theorem 5. c ( e G n ) = O (1) . The main steps in our proof are presented as lemmas.
Lemma 6.
For each edge e ∈ E ( G ) the set of all edges e e ∈ E ( e G ) for which π ( e e ) = e forms an edge cut in e G .Proof. The statement is simple for e ∈ S . In this case it is quite easy to describethe sets separated by the cut: they are the sets V ( G ) × A e, and V ( G ) × A e, where A e, and A e, are the sets of functions in { , } S whose values on e are equal to 0and 1, respectively.As for edges corresponding to the tree we have the following: an edge e ∈ E ( T )splits the tree T into two components, call them A and B . The edges of S are eitherwithin one of the components A and B , or between them. We consider two sets ofvertices:The set P consisting of pairs ( v, f ) where either v ∈ A and the sum of values of f corresponding to edges in S passing from A to B (recall that e / ∈ S ) is even, or v ∈ B and the sum of values of f corresponding to edges in S passing from A to B is odd.The set P consisting of pairs ( v, f ) where either v ∈ A and the sum of values of f corresponding to edges in S passing from A to B is odd, or v ∈ B and the sum ofvalues of f corresponding to edges in S passing from A to B is even.It is easy to check that edges connecting P and P in e G are those and only thoseedges whose projection to E ( G ) is e . emark . In [AGS11+, Lemma 3.3] it was observed that the more difficult part inthe proof of Lemma 6 follows from the easier part and a certain universality resulton graph lifts.Now we are ready to introduce an embedding F of V ( e G ) into R E ( G ) (the set ofreal-valued functions on E ( G )), which has the desired property: the distortion of F is bounded above by a universal constant if we endow R E ( G ) with its ℓ -norm.For each edge cut R ( e ) defined by the set of edges e e in e G satisfying π ( e e ) = e , wecall one of the sides of the cut R ( e ) the 0 -side , and the other side the 1 -side andintroduce a { , } -valued function h e on V ( e G ) given by h e ( x ) = 0 if x is in the 0-sideof R ( e ), and by h e ( x ) = 1 if x is in the 1-side of R ( e ).If we endow R E ( G ) with its ℓ -norm the Lipschitz constant of this embedding is1. In fact, the cuts R ( e ) are disjoint and each edge of e G is in exactly one of thecuts. Therefore || F ( x ) − F ( y ) || = 1 if x and y are adjacent vertices of e G .To estimate the Lipschitz constant of F − we consider x, y ∈ V ( e G ) and observethe following: Observation 8. If P is a path between x and y in e G , then d e G ( x, y ) ≤ length( P ) and || F ( x ) − F ( y ) || is the number of edges in the walk π ( P ) which are repeated inthe walk an odd number of times. This observation shows that we can prove the statement about the distortion if,for each pair x, y of vertices in e G we can show that a shortest xy -path P in e G issuch that sufficiently many of the edges in the walk π ( P ) are repeated only one timeand other edges are repeated at most two times.Remark 4 (on backtracking) implies that a vertex in the subgraph of G inducedby edges of π ( P ) can have degree 1 if and only if it is the projection of either thebeginning or the end of the path P . Lemma 9.
Let x, y ∈ V ( e G ) and let P be a shortest xy -path. Let I ( P ) be thesubgraph of G induced by edges of π ( P ) . Only cut edges of I ( P ) can be repeated inthe walk π ( P ) . Cut edges of I ( P ) cannot be repeated in the walk π ( P ) more thantwice.Proof. It is convenient to consider a non-simple graph N ( P ) having I ( P ) as itsunderlying simple graph and having as many parallel edges for each edge of I ( P ), asmany times the edge is repeated in π ( P ). It is easy to see that N ( P ) has an Eulertrail starting at π ( x ) and ending at π ( y ) (we just follow the projection π ( P ), usingdifferent parallel edges instead of repeating edges of the underlying graph).To prove the first statement of the lemma, we assume the contrary, that is, thereis an edge e in I ( P ) which is not a cut edge, but is repeated in π ( P ). This assumptionimplies that if we delete from N ( P ) two edges parallel to e , the result, which wedenote N ′ , will still be a connected graph. y the well-known characterization of graphs having Euler trails, degrees of allvertices of N ( P ), except possibly π ( x ) and π ( y ), are even. It is clear that thiscondition still holds for the degrees of N ′ . Using the well-known characterization ofgraphs having Euler trails again, we get that the remaining graph contains an Eulertrail which starts at π ( x ) and ends at π ( y ).We claim that the lift of this trail, if we start the lift at x , will end at y , thusgiving a shorter xy -path and leading to a contradiction.To prove the claim, we observe that our construction of the lift of G and ourdefinition of a lifted walk (see Remark 3) are such that the change in the L -coordinatein each step (when we walk along the lifted walk) is made only in one value of thecorresponding { , } -valued function on S , the choice of this coordinate dependsonly on the π -projection of the edge which we are passing, and not on the directionin which we pass it, or on the L -coordinate of the vertex we are at (this is a veryimportant property of the graph lift which we consider). Also, we need an obviousobservation that if we change some value of a { , } -valued function twice, it returnsto its original value. Hence the total change in the L -coordinate as we walk along thelift of the Euler trail of N ′ is the same as for the original Euler trail in N ( P ) (formallyspeaking, we need to replace Euler trails in N ′ and N ( P ) by the corresponding walksin the underlying simple graph I ( P )). Hence we end up at y .We can get a contradiction in the same way if we assume that some of the edgesin π ( P ) are repeated more than twice. Proving the first statement we used theassumption that e is not a cut edge only once: when we claimed that removing twocopies of e from N ( P ) we get a connected graph. For the second statement we usethe following trivial observation instead: if there is a triple of parallel edges in N ( P ),deletion of two of them does not disconnect the graph.Lemma 9 shows that to complete the proof of the theorem it remains to showthat the number of cut edges in I ( P ) (where P is a shortest xy -path) which arerepeated twice in π ( P ) cannot be much larger than the number of the remainingedges. To show this we consider two types of subgraphs in I ( P ): (i) Maximal 2-edge-connected subgraphs. Let C be the number of such subgraphs. (ii) Maximal subgraphs satisfying the conditions: (1) They are paths; (2) All in-ternal vertices of these paths (if any) have degree 2 in I ( P ); (3) All edges ofthese paths are cut edges of I ( P ). Let N be a number of such subgraphs. Observation 10.
The number of edges in each subgraph of the type (i) is at least g ( G ) . This statement is easy to see, because each such subgraph contains a cycle in G . Lemma 11. N ≤ C + 1 .Proof. We contract each maximal 2-edge-connected subgraph to a vertex, and denotethe obtained graph by D . It is clear that D is a tree, and each subgraph of type ii) is mapped into D isomorphically, and has properties (A) It is a path; (B) Allinternal vertices of this path (if any) have degree 2 in D ; (C) All edges of these pathsare cut edges of D (these properties are analogues of (1), (2), and (3) described in (ii) for D ). On the other hand, maximality can be lost. The maximality is lost inthe cases where there is a maximal 2-edge-connected subgraphs of I ( P ) incident withexactly 2 cut edges of I ( P ). Denote the number of such maximal 2-edge-connectedsubgraphs by H .Denote by N the number of maximal subgraphs in D having properties (A),(B), and (C), the previous paragraph implies that we have N − N = H .Observe that, by Remark 4 (on backtracking), all, except possibly two, of leavesof D correspond to maximal 2-edge-connected subgraphs, so we need to estimatethe number of these leaves, let us denote it by J .One of the ways to do this is to replace all paths with internal vertices of degree2 in D by edges, and denote the obtained tree by D . The number of edges in D is equal to N , and all vertices in D which are not leaves have degrees at least 3.Also D and D have the same number of leaves.So we need to estimate from below the number J of leaves in a graph with N edges and all vertices which are not leaves having degrees at least 3. Counting thesum of all degrees of D in two ways we get2 N ≥ N − J + 1) + J, or 2 J ≥ N + 3. We have 2 C ≥ J −
2) + 2 H ≥ J + H − ≥ N + H − N − Lemma 12.
Let K be a path in I ( P ) satisfying the conditions: (a) All of its internalvertices have degree in I ( P ) ; (b) Each of its edges is repeated twice in the walk π ( P ) , where P is a shortest xy -path in e G ( x, y ∈ V ( e G )) . Then the length of K is ≤ diam G .Proof. Let u, v be the ends of K . If the length of K is more than diam G , then thereis a strictly shorter uv -path K ′ in G , we are going to use this path to construct ashorter than P path in e G joining x and y . We do this in the most straightforwardway: first we modify the walk π ( P ) in the following way: each time when we walkthrough K , we walk through K ′ instead. It remains to show that if we lift this walkto e G , we get another xy -walk.In fact, this new walk clearly starts at π ( x ) and ends at π ( y ). We need onlyto check that the L -coordinate at the end of the walk will be the same as for theoriginal walk. This follows immediately from the observation that we made earlier:if we walk through two edges with the same π -projection twice, the correspondingchanges in the L -coordinate cancel each other. Since this happens for each edge ofboth K and K ′ , the L -coordinates corresponding to π ( y ) at the end of the walkswill be the same for lifts of both walks. This proves the lemma. roof of Theorem 5. We consider two cases separately:Case 1. C = 0. By Lemma 11, N = 1 in this case. We get, by Remark 4(on backtracking), that each edge of I ( P ) is used in the walk π ( P ) exactly once,therefore d e G ( x, y ) = length( P ) = || F ( x ) − F ( y ) || in this case.Case 2. C >
0. Let M be the number of cut edges in I ( P ) which are used oncein the walk π ( P ). Let M be the number of edges of I ( P ) which are in 2-edge-connected components of I ( P ). Let M be the number of cut edges of I ( P ) whichare used twice in the walk π ( P ). We have || F ( x ) − F ( y ) || = M + M . On the other hand, d e G ( x, y ) = M + M + 2 M . In addition, by Observation 10, and (2) we have M ≥ C · g ( G ) ≥ C · c · diam( G ).On the other hand, by Lemma 12, we have M ≤ N · diam( G ) ≤ C diam( G ) (if C ≥ M /M ≤ /c , and the quotient d e G ( x, y ) / || F ( x ) − F ( y ) || isbounded above by a universal constant. Remark
13 (Remark on applications to coarse embeddings) . Since ℓ admits a coarseembedding into a Hilbert space (see [Nao10, Corollary 3.1]), Theorem 5 implies thatthe graphs e G n admit uniformly coarse embeddings into a Hilbert space. Therefore,combining our Theorem 5 with a recent result of Willett [Wil11], we get more ex-amples of metric spaces with bounded geometry but without property A, admittingcoarse embeddings into a Hilbert space (first examples of this type were found in[AGS11+]). (It is worth mentioning that without the bounded geometry conditionsuch examples were known earlier [Now07].)Also, it is worth mentioning that in [Ost09] it was proved that locally finite metricspaces which do not admit coarse embeddings into a Hilbert space contain substruc-tures which are “locally expanding” (see [Ost09] for details). Our example, as wellas the example in [AGS11+], show that the converse it false, since families of graphswith constant degree ≥ References [AL06] A. Amit, N. Linial, Random lifts of graphs: edge expansion,
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