SSMALL SEPARATIONS IN VERTEX TRANSITIVEGRAPHS
MATT DEVOS AND BOJAN MOHAR
Abstract.
Let k be an integer. We prove a rough structure theo-rem for separations of order at most k in finite and infinite vertextransitive graphs. Let G = ( V, E ) be a vertex transitive graph, let A ⊆ V be a finite vertex-set with | A | ≤ | V | and |{ v ∈ V \ A : u ∼ v for some u ∈ A }| ≤ k . We show that whenever the diameter of G is at least 31( k +1) , either | A | ≤ k + k , or G has a ring-like structure(with bounded parameters), and A is efficiently contained in an interval.This theorem may be viewed as a rough characterization, generalizingan earlier result of Tindell, and has applications to the study of productsets and expansion in groups. Overview
The study of expansion in vertex transitive graphs and in groups dividesnaturally into the study of local expansion, or connectivity , and the study ofglobal expansion, or growth . The expansion properties of a group are thoseof its Cayley graphs, so vertex transitive graphs are the more general setting.Our main result, Theorem 1.9, concerns local expansion in vertex transitivegraphs, but it is also meaningful for groups, and it has some asymptotic ap-plications. Theorem 1.9 can be viewed as a rough characterization of vertextransitive graphs with small separations. It is shown that a separator oforder k in a vertex transitive graph separates a set of bounded size unlessthe graph has a ring-like structure and the separated set is essentially aninterval in this structure. This result reaches far beyond any previous re-sults on separation in vertex transitive graphs, since k does not need to bebounded in terms of the degree of the graphs. Several corollaries (1.3, 1.10,1.17, and 1.19) indicate some of diverse possibilities of applications of ourmain theorem.Graphs and groups appearing in this paper may be finite or infinite. Nev-ertheless, it is assumed that graphs are locally finite and groups are finitelygenerated.We continue with a tour of some of the important theorems in expansion. Supported in part by an NSERC Discovery Grant.Supported in part by an NSERC Discovery Grant, by the Canada Research ChairProgram, and by the ARRS, Research Program P1-0297.On leave from IMFM & FMF, Department of Mathematics, University of Ljubljana,1000 Ljubljana, Slovenia. a r X i v : . [ m a t h . C O ] O c t M. DEVOS AND B. MOHAR
Local Expansion in Groups.
This is the study of small sum sets or smallproduct sets. Let G be a (multiplicative) group and let A, B ⊆ G . The mainquestions of interest here are lower bounds on | AB | , and in the case when | AB | is small, finding the structure of the sets A and B . The first importantresult in this area was proved by Cauchy and (independently) Davenport.For every positive integer n , we let Z n = Z /n Z . Theorem 1.1 (Cauchy [6], Davenport [7]) . Let p be prime and let A, B ⊆ Z p be nonempty. Then | A + B | ≥ min { p, | A | + | B | − } . This theorem was later refined by Vosper who found the structure of all
A, B ⊆ Z p ( p prime) for which | A + B | < | A | + | B | . Before stating histheorem, note that in any finite group G we must have AB = G whenever | A | + | B | > |G| by the following pigeon hole argument: { a − g : a ∈ A }∩ B (cid:54) = ∅ for every g ∈ G . For simplicity, we have excluded this uninteresting casebelow. Theorem 1.2 (Vosper [34]) . Let p be a prime and let A, B ⊆ Z p be nonempty.If | A + B | < | A | + | B | ≤ p , then one of the following holds: (i) | A | = 1 or | B | = 1 . (ii) There exists g ∈ Z p so that B = { g − a : a ∈ Z p \ A } , and A + B = Z p \ { g } . (iii) A and B are arithmetic progressions with a common difference. Analogues of the Cauchy-Davenport theorem and Vosper’s theorem forabelian and general groups were found by Kneser [27], Kempermann [26],and recently by DeVos [8], and DeVos, Goddyn and Mohar [10].In abelian groups, we have powerful theorems which yield rough struc-tural information when a finite subset A ⊆ G satisfies | A + A | ≤ c | A | fora fixed constant c . Freiman [14] proved such a theorem when G = Z andthis has recently been extended to all abelian groups by Green and Ruzsa[16]. Despite this progress, there is still relatively little known in terms ofrough structure of sets with small product in general groups. The followingcorollary of our main theorem is a small step in this direction. Theorem 1.3.
Let G be an infinite group, let B = B − ⊆ G be a finitegenerating set containing the identity element of G , and let A ⊆ G be a finitesubset of G . If | BA | < | A | + | A | , then G has a finite normal subgroup N so that G /N is either cyclic or dihedral. Furthermore, | N | < | A | / . The proof is given towards the end of the paper in Section 6. Our resultalso applies to finite groups, but for this it requires an assumption which ismore natural in the context of graphs and will be discussed in the sequel.
Local Expansion in Graphs.
Before we begin our discussion of expan-sion in graphs, we will need to introduce some notation. If G is a graphand X ⊆ V ( G ), we let δX = { uv ∈ E ( G ) : u ∈ X and v (cid:54)∈ X } and we call EPARATIONS IN SYMMETRIC GRAPHS 3 any set of edges of this form an edge-cut . We let ∂X = { v ∈ V ( G ) \ X : uv ∈ E ( G ) for some u ∈ X } and we call ∂X the boundary of X . Similarly,if (cid:126)G is a directed graph and X ⊆ V ( (cid:126)G ) we let δ + ( X ) = { ( u, v ) ∈ E ( (cid:126)G ) : u ∈ X and v (cid:54)∈ X } and δ − ( X ) = δ + ( V ( G ) \ X ), and we let ∂ + ( X ) = { v ∈ V ( G ) \ X : ( u, v ) ∈ E ( G ) for some u ∈ X } and ∂ − ( X ) = { v ∈ V ( G ) \ X :( v, u ) ∈ E ( G ) for some u ∈ X } . Expansion in graphs is the study of the be-havior of the cardinalities of δX and ∂X . We will be particularly interestedin the case when these parameters are small. Next we introduce the typesof graphs we will be most interested in.Again we let G denote a multiplicative group. For every A ⊆ G , we definethe Cayley digraph C ay ( G , A ) to be the directed graph (without multipleedges) with vertex-set G and ( x, y ) an arc if y ∈ Ax . Using this definition,the group G has a natural (right) transitive action on V ( G ) which preservesincidence. If 1 ∈ A , and B ⊆ G , then the product set AB is the disjointunion of B and ∂ + ( B ). This observation allows us to rephrase problemsabout small product sets in groups as problems concerning sets with smallboundary in Cayley digraphs.Next we overview known bounds on the boundary of finite sets in vertextransitive graphs. These theorems are usually stated only for finite graphs,but more general versions stated below follow from the same arguments. Theorem 1.4 (Mader [28]) . If G is a connected d -regular vertex transitivegraph and ∅ (cid:54) = A ⊂ V ( G ) is finite, then | δA | ≥ d . Theorem 1.5 (Mader [29], Watkins [35]) . If G is a connected d -regularvertex transitive graph and ∅ (cid:54) = A ⊂ V ( G ) is finite and satisfies A ∪ ∂A (cid:54) = V ( G ) , then | ∂A | ≥ ( d + 1) . Theorem 1.6 (Hamidoune [20]) . If G is a connected vertex transitive di-rected graph with outdegree d and ∅ (cid:54) = A ⊂ V ( G ) is finite and satisfies A ∪ ∂ + ( A ) (cid:54) = V ( G ) , then | ∂ + ( A ) | ≥ d +12 . For the next result, recall that a partition σ (whose parts are called blocks )of vertices of a vertex transitive graph G is said to be a system of imprimi-tivity if for every automorphism ϕ of G and every block B ∈ σ , the set ϕ ( B )is another block of σ . In that case the blocks of σ are also called blocks ofimprimitivity . Having a system of imprimitivity σ , we define the quotientgraph G σ whose vertices are the blocks of imprimitivity, and two such blocks B, B (cid:48) are adjacent if there exist u ∈ B and u (cid:48) ∈ B (cid:48) that are adjacent in G .The next theorem may be viewed as a refinement of Mader’s theoremwhich gives a structural result for graphs which have small edge-cuts. Letus recall that a vertex-set B in a graph G is called a clique if every twovertices in B are adjacent. Theorem 1.7 (Tindell [32]) . Let G be a finite connected d -regular vertextransitive graph. If there exists X ⊆ V ( G ) with | X | , | V ( G ) \ X | ≥ so that | δX | = d , then one of the following holds: M. DEVOS AND B. MOHAR (i)
There is a system of imprimitivity whose blocks are cliques of order d . (ii) d = 2 (so G is a cycle). The following recent theorem of van den Heuvel and Jackson gives a roughanalogue of Tindell’s result under the assumption that G has a fairly smalledge-cut with sufficiently many vertices on either side. Theorem 1.8 (van den Heuvel, Jackson [21]) . If G is a finite connected d -regular vertex transitive graph and there exists a set S ⊆ V ( G ) with ( d +1) ≤ | S | ≤ | V ( G ) | and | δS | < ( d + 1) , then there is a block of imprimitivitythat has less than ( d + 1) vertices. Our main theorem may be viewed as a rough characterization that gen-eralizes Tindell’s result, but without any assumptions relating to the degreeof the graph and with an added assumption that the diameter of the graphis large. Before introducing the theorem we will require some further defi-nitions. For A ⊆ V ( G ), we denote by G [ A ] the subgraph of G induced onthe vertices in A . We define the diameter of A , denoted diam ( A ), as thesupremum of d ist G ( x, y ) over all x, y ∈ A . The diameter of G , denoted d iam ( G ), is defined to be d iam ( V ( G )). If A is a proper subset of V ( G ),then the depth of a vertex v in A is d ist G ( v, V ( G ) \ A ). The depth of A ,denoted d epth ( A ), is the supremum over all vertices v in A of the depth of v in A .If S is a set, a cyclic order on S is a symmetric relation ∼ so that thecorresponding graph is either a circuit, or a two-way-infinite path. The distance between two elements in S is defined to be the distance in thecorresponding graph, and an interval of S is a finite subset { s , s , . . . , s m } ⊆ S with s i ∼ s i +1 for every 1 ≤ i ≤ m −
1. A cyclic system (cid:126)σ on a graph G isa system of imprimitivity σ on V ( G ) equipped with a cyclic order (indicatedby the arrow) which is preserved by the automorphism group of G . If s, t arepositive integers, we say that G is ( s, t ) -ring-like with respect to the cyclicsystem (cid:126)σ if every block of σ has size s and any two adjacent vertices of G are in blocks which are at distance ≤ t (in the cyclic order) in (cid:126)σ .The main result of this paper is: Theorem 1.9.
Let G be a vertex transitive graph, let A ⊆ V ( G ) be a finitenon-empty set with | A | ≤ | V ( G ) | such that G [ A ∪ ∂A ] is connected. Set k = | ∂A | and assume that diam ( G ) ≥ k + 1) . Then one of the followingholds: (i) d epth ( A ) ≤ k and | A | ≤ k + k ( and G is d -regular where d ≤ k − . (ii) There exist integers s, t with st ≤ k and a cyclic system (cid:126)σ on G sothat G is ( s, t ) -ring-like, and there exists an interval J of (cid:126)σ so thatthe set Q = ∪ B ∈ J B satisfies A ⊆ Q and | Q \ A | ≤ k + k . EPARATIONS IN SYMMETRIC GRAPHS 5
This is a structure theorem giving a rough characterization of vertex tran-sitive graphs with small separations in the sense that any set A which sat-isfies (i) or (ii) must have | ∂A | bounded as a function of k . Indeed, if A satisfies (i) then | ∂A | ≤ d | A | ≤ (2 k + k )( k − ≤ k and if A satisfies(ii) then | ∂A | ≤ | ∂Q | + | Q \ A | ≤ st + k + k ≤ k + k + k .Our theorem has an immediate consequence for separations in Euleriandigraphs. Note that finite vertex transitive digraphs are always Eulerian, sothe difference only occurs in the infinite case. Let (cid:126)G be a vertex transitivedigraph, and let G be the underlying unoriented graph (which is clearlyvertex transitive). Let A ⊆ V ( (cid:126)G ) be a finite vertex-set such that 0 < | A | ≤ | V ( G ) | and G [ A ∪ ∂A ] is connected, and set k = | ∂ + ( A ) | . Let usalso assume that d iam ( G ) ≥ k + 1) . It follows from Theorem 1.6that every vertex in G has indegree and outdegree d where d ≤ k − | ∂ − ( A ) | ≤ | δ − ( A ) | = | δ + ( A ) | ≤ k (2 k −
1) and we find that | ∂A | ≤ k (in the unoriented graph G ). Thus, by the preceding theorem,either | A | ≤ k +4 k or G is ( s, t )-ring-like with st ≤ k and A is efficientlycontained in an interval. Corollary 1.10.
Let (cid:126)G be a connected vertex transitive Eulerian digraph.Let A ⊆ V ( (cid:126)G ) be a finite vertex-set such that < | A | ≤ | V ( (cid:126)G ) | and (cid:126)G [ A ∪ ∂A ] is connected, and set k = | ∂ + ( A ) | . Let us also assume that thediameter of the underlying undirected graph is at least k + 1) . Thenone of the following holds. (i) | A | ≤ k + 4 k . (ii) There exist integers s, t with st ≤ k and a cyclic system (cid:126)σ on (cid:126)G sothat (cid:126)G is ( s, t ) -ring-like and there exists an interval J of (cid:126)σ so thatthe set Q = ∪ B ∈ J B contains A and | Q \ A | ≤ k + 4 k . Interestingly, the same conclusion does not hold for (vertex transitive)digraphs which are not Eulerian. Let (cid:126)H be an orientation of the infinite 3-regular tree such that every vertex has outdegree 1 and indegree 2. Then thevertex-set B of a directed path has | ∂ + ( B ) | = 1 but B may have arbitrarilylarge size.The main notion that we use in the proof is the depth of a set. This is aconvenient parameter for our purposes, but leads us to make an assumptionon the diameter of G (to “spread out” the graph) which is likely unnecessarilystrong. As far as we know, this theorem may be true without any suchassumption. Since we work primarily with depth, the bound on d epth ( A )in (i) is the natural consequence of our arguments. To get a bound on thenumber of vertices in A for (i) we (rather naively) apply the following prettytheorem which relates | A | , | ∂A | and d iam ( A ). Theorem 1.11 (Babai and Szegedy [4]) . If G is a connected vertex tran-sitive graph and A ⊂ V ( G ) is a non-empty finite vertex-set with | A | ≤ M. DEVOS AND B. MOHAR12 | V ( G ) | , then | ∂A || A | ≥ d iam ( A ) + 1 . It appears likely that Theorem 1.9 should hold with a bound of the form | A | ≤ ck instead of | A | ≤ k (1 + o (1)) in (i). This strengthening wouldfollow from the following conjecture that the diameter in Theorem 1.11 maybe replaced by a constant multiple of the depth. Conjecture 1.12.
There exists a fixed constant c > so that in everyconnected vertex transitive graph we have | ∂A || A | ≥ c d epth ( A ) whenever A ⊆ V ( G ) is finite and < | A | ≤ | V ( G ) | . Asymptotic Expansion in Groups.
Asymptotic expansion or growth ingroups is an extensive and well studied topic. Here, instead of looking at | AB | for a pair of finite sets A, B , we consider the asymptotic behaviorof | A n | when A is a generating set. The major result in this area is thefollowing theorem of Gromov which resolved (in the affirmative) a conjectureof Milnor. Theorem 1.13 (Gromov [17]) . Let G be an infinite group, let A ⊆ G be afinite generating set, and assume further that ∈ A and { a − : a ∈ A } = A .Then the function n (cid:55)→ | A n | is bounded by a polynomial in n if and only if G has a nilpotent subgroup of finite index. In the special case that the growth is linear, the above theorem impliesthat G has a subgroup isomorphic to Z of finite index, and by a result ofFreudenthal [15] (see also Stallings [31]), this implies that G has a finitenormal subgroup N so that G /N is either cyclic or dihedral. A clear proofof this special case, which also features good explicit bounds, was obtainedby Imrich and Seifter. Theorem 1.14 (Imrich and Seifter [23]) . Let G be an infinite group, let A ⊆ G be a finite generating set, and assume further that ∈ A and { a − : a ∈ A } = A . If there exists an integer k such that k ≥ | A k | − | A k − | =: q ,then G has a cyclic subgroup of index ≤ q . In particular, G has linear growth. This result may also be obtained as a consequence of our Corollary 1.17which appears in the next section.
Asymptotic Expansion in Graphs.
Before discussing this topic, we re-quire two more definitions. For any vertex x ∈ V ( G ) and any positive integer k , we let B ( x, k ) denote the set of vertices at distance at most k from x . If G is vertex transitive, then | B ( x, k ) | = | B ( y, k ) | for every x, y ∈ V ( G ). Thefunction b : N → N given by b ( k ) = | B ( x, k ) | is called the growth function of G .The study of asymptotic expansion in graphs is the study of the behaviorof the growth function. It is easy to see that if G = C ay ( G , A ), where 1 ∈ A , EPARATIONS IN SYMMETRIC GRAPHS 7 then b ( k ) = | A k | , so this is a direct generalization of the study of expansionin groups. The following result is the major accomplishment in this areaand gives a direct generalization of Gromov’s theorem. Theorem 1.15 (Trofimov [33]) . Let G be a vertex transitive graph andassume that its growth function is bounded by a polynomial. Then thereexists a system of imprimitivity σ with finite blocks so that Aut ( G σ ) is finitelygenerated, has a nilpotent subgroup of finite index, and the stabilizer of everyvertex in G σ is finite. As before, in the case when the growth function b is bounded by a linearfunction, the structure of G can be obtained by a more elementary combi-natorial argument, as in the following result. Theorem 1.16 (Imrich and Seifter [24]) . Let G be an infinite connectedvertex transitive graph, and let b be the growth function of G . Then G hastwo ends if and only if b ( n ) is bounded by a linear function in n . Our Theorem 1.9 can be used to obtain a result similar to the above,but it also gives the following explicit lower bound on the growth of infinitevertex transitive graphs which are not ring-like.
Corollary 1.17. If G is a connected infinite vertex transitive graph anda finite vertex-set A has d epth ( A ) > | ∂A | , then G is ( s, t ) -ring-like where st ≤ | ∂A | . In particular, b ( n ) > n ( n + 1) (for every n ≥ ) if G is notring-like.Proof. The first conclusion follows directly from Theorem 1.9. Consider nowthe set A = B ( x, n ). Clearly, d epth ( A ) ≥ n +1. So, if the previous case doesnot apply, we conclude that | ∂A | ≥ n + 1, and hence b ( n + 1) = | A | + | ∂A | ≥ b ( n ) + n + 1 for every n ≥
0. This implies that b ( n ) ≥ n ( n + 1) for n ≥ (cid:3) Structural Properties.
We now turn our attention away from expansionand toward the structure of vertex transitive graphs. Next we state animportant (yet unpublished) theorem of Babai which is related to our maintheorem.
Theorem 1.18 (Babai [3]) . There exists a function f : N → N so that everyfinite vertex transitive graph G without K n as a minor satisfies one of thefollowing properties: (i) G is a vertex transitive map on the torus. (ii) G is ( f ( n ) , f ( n )) -ring-like. It appears likely to us that an inexplicit version of our theorem for finitegraphs might be obtained from Babai’s theorem (which does not give thefunction f explicitly). However, at this time we do not have a proof of this.Conversely, our theorem can be used to obtain a strengthening of Babai’stheorem with explicit values for the function f ( n ). We shall explore this in a M. DEVOS AND B. MOHAR subsequent paper [9]. Here we only state the following corollary of Theorem1.9 that may be of independent interest. This result involves the notion ofthe tree-width, whose definition is postponed until Section 6.
Corollary 1.19. If G is a connected finite vertex transitive graph and k isa positive integer, then one of the following holds. (i) G is ( s, t ) -ring-like, where st ≤ k . (ii) G has tree-width ≥ k . (iii) The degree of vertices in G is at most k − and the diameter of G is less than k + 1) . The proof is given in the last section. It is easy to see that in the firstcase of Corollary 1.19, the tree-width of G is less than k . Let us observethat in the last case of Corollary 1.19, the degree of G and the diameter of G are both bounded in terms of k . Hence, the order of G is bounded interms of k , say | V ( G ) | ≤ s ( k ). Consequently, G is ( s ( k ) , G is less than s ( k )).The key tool we use to prove our main theorem is a structural lemma onvertex transitive graphs which appears to be of independent interest. Beforestating this lemma, we require another definition. A finite subset A ⊆ V ( G )is called an ( s, t )- tube if G [ A ∪ ∂A ] is connected, and there is a partition of ∂A into { L, R } (with L, R (cid:54) = ∅ ) so that d ist G ( x, y ) ≤ s whenever x, y ∈ L or x, y ∈ R and d ist G ( x, y ) ≥ t whenever x ∈ L and y ∈ R . Any partitionsatisfying this property is called a boundary partition. Lemma 1.20 (Tube Lemma) . Let G be a vertex transitive graph. If G hasa ( k, k + 6) -tube A with boundary partition { L, R } and d epth ( V ( G ) \ A )) ≥ k + 1 , then there exists a pair of integers ( s, t ) and a cyclic system (cid:126)σ so that G is ( s, t ) -ring-like with respect to (cid:126)σ , and st ≤ min {| L | , | R |} . The proof is postponed until Section 4.This lemma is also meaningful for groups (although the assumptions aremore natural in the context of graphs): If G is a Cayley graph for a group G and G has a tube which satisfies the assumptions of the Tube Lemma, then G is ( s, t )-ring-like and it follows that G has a normal subgroup N (of size ≤ s ) so that G /N is either cyclic or dihedral.2. Uncrossing
The main tool we use in the proofs of the Tube Lemma and our maintheorem is a simple uncrossing argument. Indeed, this was the main toolused to prove Theorems 1.4, 1.5, and 1.6 as well. This argument is probablyeasiest to understand with the help of a diagram, so we introduce one inFigure 1. Here it is understood that A , A are subsets of the vertex-set ofa graph G , and the sets P, Q, S, T, U, W, X, Y, Z are defined by the diagram.For example, Q = ∂A ∩ A , X = A ∩ ( V ( G ) \ ( A ∪ ∂A )), etc. Forconvenience, we will frequently refer back to this diagram. EPARATIONS IN SYMMETRIC GRAPHS 9
P Q ST U WX Y ZA A A A Figure 1.
The diagram for the uncrossing lemma. Edgesbetween the sets are only possible where indicated.
Lemma 2.1 (Uncrossing) . Let G be a graph, let A , A ⊆ V ( G ) and let thesets P , Q , S , T , U , W , X , Y , Z be defined as in Figure 1. Then we have: (i) | ∂P | + | ∂ ( P ∪ Q ∪ S ∪ T ∪ X ) | ≤ | ∂A | + | ∂A | . (ii) | ∂S | + | ∂X | ≤ | ∂A | + | ∂A | . (iii) If | ∂A | = | ∂P | = | ∂S | = k , then | Q ∪ U | = | ∂A ∩ ( A ∪ ∂A ) | ≥ k .Proof. Let us first observe that there are no edges from
P, Q, S to X, Y, Z andno edges from
P, T, X to S, W, Z . Therefore, ∂ ( P ∪ Q ∪ S ∪ T ∪ X ) ⊆ U ∪ W ∪ Y , ∂P ⊆ Q ∪ U ∪ T , and ∂A ∩ ∂A = U . Then apply the inclusion-exclusionformula to get (i).For (ii), we similarly use the fact that ∂S ⊆ Q ∪ U ∪ W and ∂X ⊆ T ∪ U ∪ Y .To prove (iii), observe that ∂A ∩ ( A ∪ ∂A ) = Q ∪ U . Now, | ∂P | + | ∂S | ≤ | T | + 2 | Q ∪ U | + | W | = | Q ∪ U | + | ∂A | + | Q | . This implies that | Q ∪ U | + | Q | ≥ k , so | Q ∪ U | ≥ k . (cid:3) Two-Ended Graphs
The purpose of this section is to establish a theorem which gives us somedetailed structural information about vertex transitive graphs with two ends.The main tool we use is a corollary of an important theorem of Dunwoody.However, since Dunwoody’s proof is rather tricky, and we have a proof ofthis corollary which we consider to be more transparent, we have included ithere. This also has the advantage of keeping the present article entirely self-contained. Before stating the main theorem from this section, we requiresome further definitions.If G is a graph, a ray in G is a one-way-infinite path. Two rays r, s in a graph G are equivalent if for any finite set of vertices X , the (unique)component of G \ X which contains infinitely many vertices of r also contains infinitely many vertices of s . This relation is immediately seen to be anequivalence relation, and the corresponding equivalence classes are calledthe ends of the graph G . By a theorem of Hopf [22] and Halin [19], everyconnected vertex transitive graph has either one, two, or infinitely manyends. We let κ ∞ ( G ) = inf {| S | : S ⊆ V ( G ) and G \ S has ≥ } . So κ ∞ ( G ) is finite if and only if G has at least two ends.If G is a graph which is ring-like with respect to the cyclic system (cid:126)σ , thenwe say that G is q - cohesive if any two vertices of G which are in the sameblock of (cid:126)σ or in adjacent blocks of (cid:126)σ can be joined by a path of length atmost q . We are now ready to state the main result from this section. Theorem 3.1.
Let G be a connected vertex transitive graph with two ends.Then there exist integers s, t and a cyclic system (cid:126)σ so that G is ( s, t ) -ring-likeand st -cohesive with respect to (cid:126)σ , and κ ∞ ( G ) = st . An important tool used to establish this result is Corollary 3.3 below,which follows from the following strong result of Dunwoody.
Theorem 3.2 (Dunwoody [12]) . Let G be an infinite connected vertex tran-sitive graph. If there exists a finite edge-cut δX of G so that both X and V ( G ) \ X are infinite, then there exists such an edge-cut δZ with the addi-tional property that for every automorphism φ of G either Z or V ( G ) \ Z isincluded in either φ ( Z ) or φ ( V ( G ) \ Z ) . Corollary 3.3 (Dunwoody) . If G is a connected vertex transitive graphwith two ends, then there exists a cyclic system (cid:126)σ on G with finite blocks. We call a subset X of vertices a part if both X and V ( G ) \ X are infinitebut ∂X is finite. If X is a part, and (cid:15) is an end, then we say that X captures (cid:15) if every ray in (cid:15) has all but finitely many vertices in X . We call X a narrow part if | ∂X | = κ ∞ ( G ).If G is a vertex transitive graph with two ends, then every automorphism φ of G either maps each end to itself, or interchanges the two ends. Wecall automorphisms of the first type shifts and automorphisms of the secondtype reflections . Define a map s ign : Aut ( G ) → {− , } by the rule that s ign ( φ ) = 1 if φ is a shift and s ign ( φ ) = − φ is a reflection.Now we are ready to provide a self-contained proof of Dunwoody’s Corol-lary 3.3. Proof of Corollary 3.3.
Let us denote the ends of G by L and R . It fol-lows from the uncrossing lemma that whenever P, Q are narrow parts thatcapture L , then P ∩ Q and P ∪ Q are also narrow parts that also capture L . More generally, the set of narrow parts that capture L is closed underfinite intersections. Note that by vertex transitivity every vertex is con-tained in a narrow part that captures L and a narrow part that captures R . For every x ∈ V ( G ), let L ( x ) ( R ( x )) be the intersection of all narrow EPARATIONS IN SYMMETRIC GRAPHS 11 parts which contain x and capture L ( R ). We claim that L ( x ) is a nar-row part. Clearly, V ( G ) \ L ( x ) is infinite. If L ( x ) were finite, let Y bethe set of vertices at distance 1 or 2 from L ( x ). For each y ∈ Y , there isa narrow part P ( y ) that contains x , captures L , and does not contain y .Since Y is finite, the intersection T = ∩ y ∈ Y P ( y ) is also a narrow part thatcontains L ( x ) but no other point at distance ≤ | ∂ ( T \ L ( x )) | = | ∂T | − | ∂L ( x ) | < | ∂T | . Since T \ L ( x ) is also a part, thelast inequality contradicts the fact that T is a narrow part. This shows that L ( x ) is infinite.Similarly, if | ∂L ( x ) | > κ ∞ ( G ), then there exists a finite set of narrowparts containing x and capturing L with intersection T and | ∂T | > κ ∞ ( G ),a contradiction. Thus, L ( x ), and similarly R ( x ), is a narrow part.Next, define a map β R : V ( G ) × V ( G ) → Z by the rule β R ( x, y ) = | R ( x ) \ R ( y ) | − | R ( y ) \ R ( x ) | . Let x, y, z ∈ V ( G ), and define the followingvalues: a = | R ( x ) \ ( R ( y ) ∪ R ( z )) | b = | ( R ( x ) ∩ R ( y )) \ R ( z ) | c = | R ( y ) \ ( R ( z ) ∪ R ( x )) | d = | ( R ( y ) ∩ R ( z )) \ R ( x ) | e = | R ( z ) \ ( R ( y ) ∪ R ( x )) | f = | ( R ( z ) ∩ R ( x )) \ R ( y ) | Now we have that β R ( x, y ) + β R ( y, z ) = ( a + f ) − ( c + d ) + ( b + c ) − ( e + f ) = ( a + b ) − ( d + e ) = β R ( x, z ). Define β L : V ( G ) × V ( G ) → Z bythe similar rule β L ( x, y ) = | L ( x ) \ L ( y ) | − | L ( y ) \ L ( x ) | and observe that β L ( x, y ) + β L ( y, z ) = β L ( x, z ) holds. Next, define β : V ( G ) × V ( G ) → Z bysetting β ( x, y ) = β R ( x, y ) − β L ( x, y ) and note again that β ( x, y ) + β ( y, z ) = β ( x, z ). If φ ∈ Aut ( G ) and x ∈ V ( G ), then either s ign ( φ ) = 1, φ ( L ( x )) = L ( φ ( x )), and φ ( R ( x )) = R ( φ ( x )), or s ign ( φ ) = − φ ( L ( x )) = R ( φ ( x )), and φ ( R ( x )) = L ( φ ( x )). It follows that β ( x, y ) = s ign ( φ ) β ( φ ( x ) , φ ( y )) holds forevery x, y ∈ V ( G ).Now, define two vertices x, y to be equivalent if β ( x, y ) = 0. Note thatthis is an equivalence relation preserved by the automorphism group. Let σ be the corresponding system of imprimitivity. If B, B (cid:48) ∈ σ , then β ( x, x (cid:48) )has the same value for every x ∈ B and x (cid:48) ∈ B (cid:48) and we define β ( B, B (cid:48) ) tobe this value. Next, define a relation on σ as follows. For any block B ∈ σ ,there is a unique block B (cid:48) for which β ( B, B (cid:48) ) is minimally positive. Include(
B, B (cid:48) ) in our relation. It follows immediately that this relation imposes acyclic order which is preserved by any automorphism of the graph, so wehave a cyclic system (cid:126)σ as desired.It remains to show that the blocks of σ are finite. Let F ( x ) = ∂L ( x ) ∪ ∂R ( x ) ∪ ( V ( G ) \ ( ∂L ( x ) ∪ ∂R ( x ))) . Figure 2.
A vertex transitive graph of Type 2Observe that F ( x ) is finite. Set h = max { d ist ( x, z ) | z ∈ F ( x ) } . Notethat h is independent of x ∈ V ( G ) since G is vertex transitive and sinceevery automorphism φ of G maps L ( x ) and R ( x ) onto L ( φ ( x )) and R ( φ ( x ))(possibly interchanging L and R ). If d ist ( x, y ) > h , then it is easy to seethat either y ∈ L ( x ) and x ∈ R ( y ), or y ∈ R ( x ) and x ∈ L ( y ). Assuming theformer, L ( x ) is a narrow part containing y and capturing L , so L ( y ) ⊆ L ( x ).Since d ist ( x, y ) > h , we have d ist ( ∂L ( x ) , y ) > h , hence ∂L ( y ) ⊆ L ( x ).Thus β L ( x, y ) = | L ( x ) \ L ( y ) | ≥ | ∂L ( y ) | > . Similarly, since x ∈ R ( y ), we conclude that R ( x ) ∪ ∂R ( x ) ⊆ R ( y ), andconsequently β R ( x, y ) = −| R ( y ) \ R ( x ) | ≤ −| ∂R ( x ) | < . Finally, this yields that β ( x, y ) (cid:54) = 0 and shows that the blocks of σ are finite.This completes the proof. (cid:3) If G is a vertex transitive graph with two ends, then we may define arelation ∼ on V ( G ) by the rule x ∼ y if there exists a shift φ ∈ Aut ( G )with φ ( x ) = y . It is immediate from the definitions that ∼ is an equivalencerelation preserved by Aut ( G ), and we let τ denote the corresponding systemof imprimitivity. Since the product of two reflections is a shift, | τ | ≤
2. Wedefine G to be Type i if | τ | = i . Graphs of Type 1 will be easiest to workwith, since in this case we have shifts taking any vertex to any other vertex.If G is a graph of Type 2, then we view τ as a (not necessarily proper)2-coloring of the vertices. In this case, every shift fixes both color classes,and every reflection interchanges them. An example of a Type 2 graph isillustrated in Figure 2.If X, Y are disjoint subsets of V ( G ), we say that X and Y are neighborly if every point in X has a neighbor in Y and every point in Y has a neighborin X . We say that G is tightly ( s, t ) -ring-like with respect to (cid:126)σ if G is ( s, t )-ring-like with respect to (cid:126)σ , and further, every pair of blocks in (cid:126)σ at distance t are neighborly. Lemma 3.4. If G is a connected vertex transitive graph with two ends, thenthere exist integers s, t , and a cyclic system (cid:126)σ so that G is tightly ( s, t ) -ring-like with respect to (cid:126)σ .Proof. By Corollary 3.3 we may choose a cyclic system (cid:126)σ where σ = { X i : i ∈ Z } and the cyclic order is . . . , X − , X , X , . . . . Set s to be the size of a EPARATIONS IN SYMMETRIC GRAPHS 13 block of σ and t to be the largest integer so that there exist adjacent verticeswhich lie in blocks at distance t . Then, G is ( s, t )-ring-like with respect to (cid:126)σ .First suppose that G is Type 1 and choose i ∈ Z so that E [ X i , X i + t ] (cid:54) = ∅ .Then every point in X i must have a neighbor in X i + t since there exists ashift taking any point in X i to any other point in this block, and such amap must fix X t + i . Similarly, every point in X i + t has a neighbor in X i , so X i and X i + t are neighborly. For every j , there exists a shift which sends X i to X j , so X j and X j + t are also neighborly. It follows from this that G istightly ( s, t )-ring-like with respect to σ .Thus, we may assume that G is of Type 2, and we let τ = { Y , Y } be thecorresponding system of imprimitivity. If σ is not a refinement of τ , then { X i ∩ Y j : i ∈ Z and j ∈ { , }} is a system of imprimitivity. For Z j = X j ∩ Y , Z j +1 = X j ∩ Y ( j ∈ Z ), the cyclic ordering . . . , Z − , Z − , Z , Z , Z , . . . is preserved by Aut ( G ). Thus, by possibly adjusting (cid:126)σ and s and t , we mayassume that σ is a refinement of τ . In particular, for every x, y ∈ X i thereexists an automorphism sending x to y which fixes every block of (cid:126)σ (sinceevery shift sending x to y has this property). So X i and X j are neighborlywhenever E [ X i , X j ] (cid:54) = ∅ .Note that we may modify the cyclic order on σ by “shifting the even blocks2 k steps to the right”, replacing X i by X i − k for every i ∈ Z to obtain anew cyclic ordering which is preserved by Aut ( G ). Set t = sup { i ∈ Z : E [ X , X i ] (cid:54) = ∅} (if such i ∈ Z does not exist, let t = 0), set t − = min { j ∈ Z + 1 : E [ X , X j ] (cid:54) = ∅} and set t +1 = max { j ∈ Z + 1 : E [ X , X j ] (cid:54) = ∅} .Since there must be a block with odd index joined to X , t +1 and t − bothexist. By shifting even blocks, we may further assume that either t +1 = − t − or t +1 = 2 − t − . If t ≥ t +1 , then t = t and E [ X i , X i + t ] (cid:54) = ∅ for every i ∈ Z so X i and X i + t are neighborly for every i ∈ Z and we are done.Similarly, if t +1 = − t − > t , then t = t +1 and X i and X i + t are neighborlyfor every i ∈ Z and we are done. The only remaining possibility is that t = t +1 = 2 − t − > t . In this case, set σ (cid:48) = { X i ∪ X i +1 : i ∈ Z } . Then σ (cid:48) isa system of imprimitivity, . . . , X − ∪ X − , X ∪ X , X ∪ X , . . . is a cyclicorder preserved by Aut ( G ), and by construction, G is tightly (2 s, t − )-ring-like with respect to σ (cid:48) and this ordering. (cid:3) Lemma 3.5. If G is an infinite connected vertex transitive graph that istightly ( s, t ) -ring-like, then κ ∞ ( G ) = st .Proof. It is immediate that κ ∞ ( G ) ≤ st as the removal of t consecutiveblocks of size s leaves a graph with at least two infinite components. Thus,it suffices to show that st ≤ κ ∞ ( G ).Assume that G is tightly ( s, t )-ring-like with respect to (cid:126)σ where σ = { X i : i ∈ Z } and the cyclic order is given by . . . , X − , X , X , . . . . Next, choose A ⊆ V ( G ) with A and V ( G ) \ A infinite so that(i) | ∂A | is minimum;(ii) T = { y ∈ V ( G ) \ A : σ y ∩ A (cid:54) = ∅} is minimal subject to (i), where σ y denotes the block of σ containing y . It follows from our assumptions that | ∂A | = κ ∞ ( G ) is finite. Further,since there is a fixed upper bound on the maximum distance between twovertices in the same block of σ , the set T is finite. Suppose (for a con-tradiction) that there exist points x, y in the same block of σ and a shift φ ∈ Aut ( G ) so that x ∈ A , y (cid:54)∈ A , and so that φ ( x ) = y . Then φ must fixevery block of σ , so the symmetric difference of A and φ ( A ) is finite. Byuncrossing (Lemma 2.1), we have | ∂ ( A ∩ φ ( A )) | + | ∂ ( A ∪ φ ( A )) | ≤ | ∂A | . Butthen it follows from (i) that | ∂ ( A ∪ φ ( A )) | = | ∂A | and we see that A ∪ φ ( A )contradicts our choice of A for (ii). Thus, no such x, y, φ can exist.Now suppose that there are shifts taking any point in a block of σ toany other point in this block. It then follows from the above argument thatboth A and ∂A are unions of blocks of σ . Let Q k = ∪ i ∈ Z X it + k for every0 ≤ k ≤ t −
1. Then Q k ∩ ∂A must include a block of σ for every 0 ≤ k ≤ t − κ ∞ ( G ) = | ∂A | ≥ st as desired.Thus, we may assume that there exist x, y ∈ X so that no shift maps x to y . So, G is Type 2, and setting τ = { Y , Y } to be the corresponding2-coloring, we find that σ (cid:48) = { X i ∩ Y j : i ∈ Z and j ∈ { , } } is a properrefinement of σ and τ . Furthermore, it follows from our earlier analysis thatboth A and ∂A are unions of blocks of σ (cid:48) . It follows from the assumption that G is tightly ( s, t )-ring-like that either X ∩ Y j and X t ∩ Y j are neighborly for j = 1 , X ∩ Y j and X t ∩ Y l are neighborly whenever { j, l } = { , } .In the former case, setting Q jk = ∪ i ∈ Z ( X it + k ∩ Y j ) for 0 ≤ k ≤ t − j = 1 , ∂A ∩ Q jk includes a block of σ (cid:48) for every 0 ≤ k ≤ t − j = 1 , κ ∞ ( G ) = | ∂A | ≥ st as desired. In the latter case, setting Q jk = ∪ i ∈ Z ( X it + k ∩ Y ( j + i ) (mod 2) ) for 0 ≤ k ≤ t − j = 1 , ∂A ∩ Q jk includes a block of σ (cid:48) for every 0 ≤ k ≤ t − j = 1 ,
2. Again,this implies that κ ∞ ( G ) = | ∂A | ≥ st and we are finished. (cid:3) In the next proof we will need a simple lema about short paths in finitegraphs.
Lemma 3.6.
Let H be a finite connected graph, possibly with multiple edges,and let Q , . . . , Q h be pairwise disjoint cycles in H . For every x, y ∈ V ( H ) ,there exists an ( x, y ) -path P in H such that for every i (1 ≤ i ≤ h ) theintersection P ∩ Q i is either empty or a segment of Q i containing at mosthalf of the edges of Q i .Proof. Let P be an ( x, y )-path. We say that the cycle Q i is badly traversed by P if P ∩ Q i is not as claimed. Suppose now that Q i is badly traversed,and let U = V ( P ∩ Q i ). Let a, b be the first and the last vertex, respectively,on P that belongs to Q i . If we replace the ( a, b )-segment of P by a shortest( a, b )-segment on Q i , we get another ( x, y )-path P (cid:48) . Clearly, P (cid:48) does notintroduce any new badly traversed cycles among Q , . . . , Q h , and repairsbad traversing of Q i . This procedure thus leads to an ( x, y )-path which isas claimed. (cid:3) EPARATIONS IN SYMMETRIC GRAPHS 15
Lemma 3.7. If G is an infinite connected vertex transitive graph that istightly ( s, t ) -ring-like with respect to (cid:126)σ , then it is st -cohesive with respectto (cid:126)σ .Proof. Assume that σ has blocks { X i : i ∈ Z } and the cyclic order in (cid:126)σ isgiven by . . . , X − , X , X , . . . . Let x ∈ X and y ∈ X − ∪ X ∪ X . Ourgoal is to prove that d ist G ( x , y ) ≤ st .Let x ∈ X t be a neighbor of x , and let x ∈ X t be a neighbor of x .Since shifts of G have only one or two orbits, it is possible to chose x suchthat x and x are in the same orbit, i.e., there is a shift α mapping x to x . For i ∈ Z , let x i = α i ( x ) and let x i +1 = α i ( x ). Note that vertices . . . , x − , x − , x , x , x , x , . . . form a two-way-infinite path P in G that ispreserved under the action of α .Let H be the quotient graph of G under the action of α . More precisely,vertices of H are the orbits of the action of α on V ( G ), and two of themare adjacent if there is an edge in G joining the two orbits. Since α is anautomorphism of G such that α i has no fixed points if i ∈ Z \ { } , G is acover of H . Let x and y be the vertices of H that are orbits of x and y ,respectively. Since G is connected, there is a path Q from y to x in H . Since | V ( H ) | = 2 st , it is immediate that | V ( Q ) | ≤ st .By the unique lifting property of paths in covering spaces, Q can be liftedto a path ˜ Q in G joining y and some vertex x (cid:48) in the same α -orbit as x .Note that x (cid:48) ∈ X mt ( m ∈ Z ), so x can be reached from x (cid:48) by using 2 | m | edges on the path P . This shows that d ist G ( x , y ) ≤ | V ( Q ) | − | m | .In order to show that the last inequality above implies that G is 2 st -cohesive, we need to show that | V ( Q ) | + 2 | m | − ≤ st . We shall useLemma 3.6, so we first define cycles Q j covering all vertices in H . Theedges between X i and X i + t form a bipartite graph. Each component ofthis bipartite graph is a regular bipartite graph of positive degree since G isneighborly ( s, t )-ring-like. By K¨onig’s theorem, there is a perfect matching M i between X i and X i + t . We choose such perfect matchings arbitrarily forall i = 0 , , . . . , t −
1. The projection of all matchings M , M , . . . , M t − into H gives rise to a collection of disjoint cycles Q j ( j ∈ J ) covering allvertices of H . Also note that all these cycles are of even length (possiblylength 2). Let us now assume that the ( x, y )-path Q satisfies the conclusionof Lemma 3.6. If Q contains half of the edges of some cycle Q j , in whichcase we will say that the cycle Q j is problematic , then we have the freedomto choose one or the other segment of Q j to be included in Q . Note that thedefinition of the cycles Q j implies the following property: if v ∈ V ( Q j ) and vv , vv are the two edges on Q j incident with v , then for every lift of thepath v vv to a path ˜ v ˜ v ˜ v in G , if ˜ v ∈ X i , then one of the vertices ˜ v , ˜ v isin X i − t and the other one is in X i + t . This property and freedom to chooseeither half of problematic cycles Q j enables us to assume the following. Let Q = u u u . . . u r (where u = y and u r = x ) and let ˜ Q = ˜ u ˜ u ˜ u . . . ˜ u r . (*) If a cycle Q j ( j ∈ J ) sucks and if u i ( i ≥
1) is the first vertex of Q on Q j , then the (cid:126)σ -distance of ˜ u i − and ˜ u i +1 is at most t . Similarly,if the cycle containing u sucks, then ˜ u ∈ X l , where | l | ≤ t .Now, let k be the number of problematic cycles Q j . Then | V ( Q ) | ≤ ( | V ( H ) | + k ) = st + k . Property (*) guarantees that for every problematiccycle Q j (except possibly the first one) we save one for the backtracking on P , i.e. 2 | m | ≤ | V ( Q ) | − − ( k − d ist G ( x , y ) ≤ | V ( Q ) | − | m | ≤ | V ( ˜ Q ) | − k − ≤ st − . This completes the proof. (cid:3)
Proof of Theorem 3.1.
This is an immediate consequence of Lemmas 3.4,3.5, and 3.7. (cid:3) The Tube Lemma
The goal of this section is to prove (a slight strengthening of) the TubeLemma 1.20. We begin by proving a lemma similar in spirit, but for vertexcuts where all points in the cut are close to the same vertex.
Lemma 4.1.
Let X ⊆ V ( G ) be a finite vertex-set in a vertex transitivegraph G and assume that there exists y ∈ V ( G ) so that d ist G ( y, x ) ≤ k forevery x ∈ X . If there is a finite component H of G \ X with d epth ( V ( H )) ≥ k + 2 , then d epth ( V ( H (cid:48) )) < k + 2 for every other component H (cid:48) of G \ X .Proof. Let X be a minimal counterexample to the lemma. Choose a finitecomponent of G \ X with vertex-set A so that d epth ( A ) ≥ k + 2, andlet B be the vertex-set of another (finite or infinite) component of G \ X with d epth ( B ) ≥ k + 2. We may assume that | A | ≤ | B | . Next choosea point z ∈ A with depth ≥ k + 2 and choose an automorphism φ with φ ( y ) = z . Since A contains the ball of radius k + 1 around z , we have φ ( X ) ⊆ A and A \ ( φ ( A ) ∪ φ ( X )) (cid:54) = ∅ . It follows from this that S = φ ( A ) \ ( A ∪ X ) (cid:54) = ∅ . Furthermore, by our assumption on | A | , we have T = V ( G ) \ ( A ∪ X ∪ φ ( A ) ∪ φ ( X )) (cid:54) = ∅ . Now, set X (cid:48) = X ∩ φ ( A ) and X (cid:48)(cid:48) = X \ φ ( A ). Then X (cid:48) , X (cid:48)(cid:48) are vertex cuts which separate S and T (respectively) from the rest of the vertices. Since S ⊂ φ ( A ) and A is finite,we have | S | < | A | ≤ | B | . This implies that B ⊆ T is a component of G \ X (cid:48)(cid:48) of depth ≥ k + 2. Another component of G \ X (cid:48)(cid:48) has vertex-set A ∪ φ ( A ). Itis finite and has depth ≥ k + 2 as well. Therefore, X (cid:48)(cid:48) contradicts our choiceof X as a minimum counterexample. (cid:3) For i = 1 , A i be a tube with boundary partition { L i , R i } and let P, Q, S, T, U, W, X, Y, Z be the sets indicated by the diagram in Figure 1.We say that A and A merge if P, S, X, Z (cid:54) = ∅ , { Q, Y } = { L , R } and { T, W } = { L , R } . Note that these conditions imply that U = ∅ ; seeFigure 3 for intuition.The following lemma will be used to guarantee that tubes merge. EPARATIONS IN SYMMETRIC GRAPHS 17
X P QZ Y T S W ZR R L L Figure 3.
Merging two overlapping tubes into a bigger one.
Lemma 4.2.
Let G be a connected vertex transitive graph, and for i =1 , let A i be a ( k, k + 2) -tube in G with boundary partition { L i , R i } andd epth ( V ( G ) \ ( A i ∪ ∂A i )) ≥ k + 2 . Let P, Q, S, T, U, W, X, Y, Z be the setsindicated in Figure 1. If
P, S, X, Z (cid:54) = ∅ and d ist ( ∂A , ∂A ) ≥ k +12 , then A and A merge.Proof. It follows from the assumption d ist ( ∂A , ∂A ) ≥ k +12 that U = ∅ . The sets T and Q cannot be empty since G [ A ∪ ∂A ] and G [ A ∪ ∂A ] are connected and P, S, X (cid:54) = ∅ . Suppose (for a contradiction) that W = ∅ . Then Y (cid:54) = ∅ since Z (cid:54) = ∅ and G is connected. Furthermore d ist ( Q, Y ) ≥ d ist ( Q, ∂A ) + d ist ( ∂A , Y ) ≥ k + 1 and it follows that { Q, Y } = { L , R } . But then applying Lemma 4.1 to either L or R gives us a contradiction. Thus W (cid:54) = ∅ and similarly Y (cid:54) = ∅ . Again,we have d ist ( Q, Y ) ≥ d ist ( Q, ∂A ) + d ist ( ∂A , Y ) ≥ k + 1 and simi-larly d ist ( T, W ) ≥ d ist ( T, ∂A ) + d ist ( ∂A , W ) ≥ k + 1. It follows that { Q, Y } = { L , R } and { T, W } = { L , R } as required. (cid:3) The key ingredient in the proof of our tube lemma for finite graphs is theconstruction of a certain graph cover. We then use this cover together withCorollary 3.3 to obtain the desired structure. Our construction is based onvoltage assignments, and the reader is referred to Gross and Tucker [18] fora good introduction to this area. Some further notation and definitions willbe introduced in the proof of the Tube Lemma.
Lemma 4.3.
Let G be a vertex transitive graph. If G has a ( k, k + 6) -tube A with boundary partition { L, R } and d epth ( V ( G ) \ ( A ∪ ∂A )) ≥ k + 2 ,then there exists a pair of integers ( s, t ) and a cyclic system (cid:126)σ so that G is ( s, t ) -ring-like and st -cohesive with respect to (cid:126)σ , and st ≤ min {| L | , | R |} .Proof. Choose a ( k, k + 6)-tube A with d epth ( V ( G ) \ ( A ∪ ∂A )) ≥ k + 2and boundary partition { L, R } so that:(i) min {| L | , | R |} is as small as possible,(ii) | L | + | R | is as small as possible subject to (i),(iii) | A | is as small as possible subject to (i) and (ii).It suffices to show that G is ( s, t )-ring-like where st ≤ min {| L | , | R |} . Theproof consists of a series of four claims numbered (0)–(3), followed by a splitinto two cases, depending on wether G is finite or infinite. (0) Every vertex in ∂A has a neighbor in V ( G ) \ ( A ∪ ∂A ). Suppose (for a contradiction) that x ∈ ∂A has no such neighbor. If { x } (cid:54) = L and { x } (cid:54) = R then A ∪ { x } contradicts our choice of A for (i) or(ii). But if { x } = L or { x } = R , then ∂ ( A ∪ { x } ) is included in either R or L , and applying Lemma 4.1 to this set gives a contradiction. (1) If G is finite, then G \ E [ A, L ] and G \ E [ A, R ] are connected, and | A | ≤ | V ( G ) \ ( A ∪ ∂A ) | .Suppose that G \ E [ A, L ] is not connected, let B be the vertex-set of acomponent of this graph with B ∩ A = ∅ and set L (cid:48) = B ∩ L and B (cid:48) = B \ L . It is an immediate consequence of Lemma 4.1 (applied to L (cid:48) ) that d epth ( B (cid:48) ) ≤ k + 1. It then follows from the same lemma (applied to R )that L (cid:48) (cid:54) = L . But then, A ∪ B is a tube which contradicts our choice of A for (i) or (ii). Thus G \ E [ A, L ] is connected, and by a similar argument G \ E [ A, R ] is also connected. It follows from this that there is a componentof G \ ( A ∪ L ∪ R ) with vertex-set C and L ∩ ∂C (cid:54) = ∅ (cid:54) = R ∩ ∂C . Since G is finite, C is a ( k, k + 6)-tube and by assumption (iii), we conclude that | A | ≤ | C | , so | A | ≤ | V ( G ) \ ( A ∪ ∂A ) | as desired. (2) If φ ∈ Aut ( G ) satisfies A ∩ φ ( A ) (cid:54) = ∅ and d ist ( ∂A, ∂φ ( A )) ≥ k +12 , then A and φ ( A ) merge.Set A = A and A = φ ( A ) and define the sets P, Q, S, T, U, W, X, Y, Z asin Figure 1. It follows immediately from our assumptions that U = ∅ andthat P, T ∪ X, Q ∪ S (cid:54) = ∅ . If S = ∅ , then Q (cid:54) = ∅ , but every point in this set hasall its neighbors in P ∪ Q = A ∪ ∂A and we have a contradiction to claim (0).Thus S (cid:54) = ∅ and similarly X (cid:54) = ∅ . If Z = ∅ then G is finite, and by (0) we have W = Y = ∅ . But then | V ( G ) \ ( A ∪ ∂A ) | = | S | < | P | + | Q | + | S | = | A | = | A | contradicting (1). Thus Z (cid:54) = ∅ . Now Lemma 4.2 shows that A and φ ( A )merge, as claimed.Choose a shortest path D in G [ A ∪ ∂A ] from L to R . Let v − be the endof D in L , let v be the end of D in R , and let r ∈ Z be such that the lengthof D is either 2 r or 2 r + 1. Note that by our assumptions, r ≥ (cid:100) k +52 (cid:101) .Choose a vertex v in “the middle of” D , i.e., r ≤ d ist ( v , v i ) ≤ r + 1 for i ∈ {− , } , and choose vertices v − , v ∈ V ( D ) at distance (cid:100) k +52 (cid:101) from v so that v − lies on the subpath of D from v to v − and v lies on thesubpath from v to v . (3) Let φ be an automorphism of G and assume that either d ist ( φ ( v ) , v − ) ≤ d ist ( φ ( v ) , v ) ≤
1. Then d ist ( ∂A, ∂φ ( A )) ≥ k +12 , and A and φ ( A )merge. EPARATIONS IN SYMMETRIC GRAPHS 19
We give the proof in the case d ist ( φ ( v ) , v − ) ≤
1; the other case followsby a similar argument. Let y ∈ L and x ∈ ∂φ ( A ). Then we have d ist ( y, x ) ≥ d ist ( φ ( v ) , x ) − d ist ( φ ( v ) , v − ) − d ist ( v − , y ) ≥ r − ( r − k +52 + 2) − k = k +12 . Next let y ∈ R . Then any path from φ ( v ) to y which does not containa point in L has length ≥ r + k +52 − r + k +32 and any such pathwhich does contain a point in L has length ≥ d ist ( φ ( v ) , L ) + d ist ( L, R ) ≥ ( r − k +62 −
1) + (3 k + 6) ≥ r + k +42 . It follows that d ist ( φ ( v ) , y ) ≥ r + k +32 .Thus, for every x ∈ ∂φ ( A ) we have d ist ( x, y ) ≥ d ist ( φ ( v ) , y ) − d ist ( φ ( v ) , x ) ≥ ( r + k +32 ) − ( r + 1 + k )= k +12 Thus we have that d ist ( ∂A, ∂φ ( A )) ≥ k +12 and by (2) we find that A and φ ( A ) merge. Case 1: G is infinite.We shall construct a sequence ( φ i , S i ) where φ i ∈ Aut ( G ) and S i ∈{ φ i ( L ) , φ i ( R ) } recursively by the following procedure. Set ( φ − , S − ) =(1 , L ) and ( φ , S ) = (1 , R ). For i ≥ φ i ∈ Aut ( G ) and S i ∈{ φ i ( L ) , φ i ( R ) } so that the following properties are satisfied (it follows fromclaim (3) that such a choice is possible):(i) φ i ( A ) merges with φ i − ( A ).(ii) S i − ⊆ φ i ( A ).(iii) S i ∩ φ i − ( A ) = ∅ .(iv) d ist ( ∂φ i ( A ) , ∂φ i − ( A )) ≥ k +12 .For every i ≥
0, define X i = ∪ ij =0 φ j ( A ). We now prove by induction that X i is a ( k, k + 2)-tube with boundary partition { S i − , S i } for every i ≥ X = A and { S − , S } = { L, R } , so this is true for i = 0. For the inductive step, suppose that thisholds for all values less than i . If d ist ( S i , S i − ) ≤ k + 1, then there is avertex at distance ≤ (cid:100) k +12 (cid:101) from every point in ∂X i and since d epth ( X i ) ≥ d epth ( X i − ) ≥ d epth ( X ) ≥ k +62 ≥ (cid:100) k +52 (cid:101) we have a contradiction toLemma 4.1 (with (cid:100) k +12 (cid:101) playing the role of k in the lemma). But since X i − is a ( k, k + 2)-tube and d ist ( ∂φ i ( A ) , ∂X i − ) ≥ k +12 , so by applyingLemma 4.2 to the tubes X i − and φ i ( A ) we conclude that they merge. Itfollows that X i is a ( k, k + 2)-tube with boundary partition { S i − , S i } .A straightforward inductive argument now shows that the graph G hastwo ends. Furthermore, L and R are vertex cuts that separate the vertex-set into two sets of infinite size. Thus, by Theorem 3.1 we find that G is ( s, t )-ring like and 2 st -cohesive with respect to some cyclic system (cid:126)σ where st ≤ min {| L | , | R |} , as desired. Case 2: G is finite.Let us first define necessary notation for applying voltage graph construc-tion. For every graph G , we define A ( G ) = { ( u, v ) ∈ V ( G ) × V ( G ) : u and v are adjacent in G } and we call the members of A ( G ) arcs . We call a map µ : A ( G ) → Z a voltage map if µ ( u, v ) = − µ ( v, u ) for every ( u, v ) ∈ A ( G ). (This notionextends naturally to general groups, but we have restricted our attention to Z for simplicity.) For every graph G and voltage map µ , we define a graph C ( G, µ ) as follows: the vertex-set is V ( G ) × Z , and vertices ( u, i ), ( v, j ) areadjacent if uv ∈ E ( G ) and j − i = µ ( u, v ). The map π : V ( G ) × Z → V ( G )given by π ( v, g ) = v is then a covering map, so C ( G, µ ) is a cover of G .If µ, µ (cid:48) are voltage maps on G , we say that a mapping Ψ : V ( C ( G, µ )) → V ( C ( G, µ (cid:48) )) preserves π if π ◦ Ψ = π . Note that for every integer j the mapΨ j : V ( G ) × Z → V ( G ) × Z given by Ψ j ( u, i ) = ( u, i + j ) is an automorphismof C ( G, µ ) which preserves π . For every S ⊆ V ( G ) and m ∈ Z let δ mS : A ( G ) → Z be the map given by the rule δ mS ( u, v ) = m if u ∈ S and v (cid:54)∈ S − m if u (cid:54)∈ S and v ∈ S µ, µ (cid:48) : A ( G ) → Z are elementary equivalent if either µ (cid:48) = − µ or µ (cid:48) = µ + δ mS for some S ⊆ V ( G ) and m ∈ Z . Wesay that µ and µ (cid:48) are equivalent and write µ ∼ = µ (cid:48) if there is a sequence µ = µ , µ , . . . , µ n = µ (cid:48) of voltage maps on G with µ i elementary equivalentto µ i +1 for every 0 ≤ i ≤ n −
1. It is straightforward to verify that whenever µ ∼ = µ (cid:48) , there exists a bijection from C ( G, µ ) to C ( G, µ (cid:48) ) which preserves π .By possibly switching L and R , we may assume that | L | ≤ | R | . For every φ ∈ Aut ( G ), define the voltage map µ φ : A ( G ) → Z by the following rule: µ φ ( u, v ) = u ∈ φ ( L ) and v ∈ φ ( A ) − v ∈ φ ( L ) and u ∈ φ ( A )0 otherwiseWe shall use the following properties of voltage maps assigned to tubes.First of all, if we interchange the roles of L and R in the definition of µ φ , weget another voltage map µ (cid:48) φ that is equivalent to µ φ since µ (cid:48) φ = µ φ + δ − φ ( A ) .Second, if tubes φ ( A ) and φ ( A ) merge, then µ φ ∼ = µ φ .Let φ , φ ∈ Aut ( G ) satisfy d ist ( φ ( v ) , φ ( v )) ≤
2, where v is the ver-tex selected before claim (3). Then choose u ∈ V ( G ) so that d ist ( u, φ i ( v )) ≤ i = 1 , ψ ∈ Aut ( G ) so that ψ ( v − ) = u (where v − is asdefined before claim (3)). It follows from claim (3) that ψ ( A ) and φ i ( A ) EPARATIONS IN SYMMETRIC GRAPHS 21 merge for i = 1 ,
2. It follows from this that µ φ ∼ = µ ψ ∼ = µ φ . We concludethat µ φ ∼ = µ for every φ ∈ Aut ( G ).Let ˜ G = C ( G, µ ) and for every i ∈ Z let A i = { ( v, i ) : v ∈ V ( G ) } . Byclaim (1) we have that G \ E [ A, L ] is connected, and it follows that { G [ A i ] : i ∈ Z } is the set of components of ˜ G \{ uv ∈ E ( ˜ G ) : π ( u ) ∈ L and π ( v ) ∈ A } .It follows that ˜ G has two ends, and that κ ∞ ( ˜ G ) ≤ | L | .Let G = Aut ( G ) and let ˜ G = { φ ∈ Aut ( ˜ G ) : φ preserves π } . Then definethe map ν : ˜ G → G by the rule ν ( ψ ) v = π ( ψ ( v, ν ( ψ ) is simply thenatural projection of ψ ). The following diagram now shows the actions ofthe group G on the graph G and of ˜ G on ˜ G = C ( G, µ ).˜ G ˜ G G G (cid:45)(cid:63) ν (cid:63) π (cid:45) Next we shall prove that the map ν is onto. Let φ ∈ G and define˜ φ : V ( G ) × Z → V ( G ) × Z by the rule ˜ φ ( v, i ) = ( φ ( v ) , i ). Then ˜ φ is anisomorphism from C ( G, µ ) to C ( G, µ φ − ). Since µ ∼ = µ φ − we may choosean isomorphism ψ : C ( G, µ φ − ) → C ( G, µ ) which preserves π . With thesedefinitions in place, we now have the following commuting diagram. C ( G, µ ) C ( G, µ φ − ) C ( G, µ ) G G G (cid:45) ˆ φ (cid:63) π (cid:45) ψ (cid:63) π (cid:63) π (cid:45) φ (cid:45) Thus, we have that ˜ φ ◦ ψ ∈ ˜ G so ν is onto. Thus ˜ G is vertex transitive,and by Theorem 3.1 we have that ˜ G is ( s, t )-ring-like and 2 st -cohesive withrespect to some cyclic system (cid:126)σ where st = κ ∞ ( ˜ G ) ≤ | L | . Since ˜ G is aregular cover, τ = { π ( X ) : X ∈ σ } is a partition of G . Since ν is onto, weconclude that τ is a system of imprimitivity on G . Now, τ inherits a cyclicordering (cid:126)τ from (cid:126)σ , and it follows that G is ( s, t )-ring-like and 2 st -cohesivewith respect to (cid:126)τ . Since st ≤ | L | , this completes the proof. (cid:3) Main Theorem
The purpose of this section is to prove our main result, Theorem 1.9. Webegin by establishing a lemma on the structure of separations in ring-likegraphs.
Lemma 5.1.
Let G be a vertex transitive graph which is ( s, t ) -ring-like and st -cohesive with respect to (cid:126)σ . Let A ⊆ V ( G ) and assume that G [ A ∪ ∂A ] isconnected, | A | ≤ | V ( G ) | and | ∂A | = k . Then there exists an interval J of (cid:126)σ so that the set Q = ∪ B ∈ J B satisfies A ⊆ Q and | Q \ A | ≤ s t k + 2 stk . Proof.
Let J be the set of all B ∈ (cid:126)σ with the property that A ∩ B (cid:54) = ∅ and A ∩ ( B ∪ B (cid:48) ) (cid:54) = B ∪ B (cid:48) for some B (cid:48) ∈ (cid:126)σ with B (cid:48) at (cid:126)σ -distance at most 1 from B . Consider a block B ∈ J . Let a ∈ A ∩ B and b ∈ ( B ∪ B (cid:48) ) \ A . Since G is 2 st -cohesive with respect to (cid:126)σ , there is a path in G from a to b of length ≤ st . This path contains at least one vertex in ∂A . Since this path startsin B and ends in B ∪ B (cid:48) , its maximum (cid:126)σ -distance from B ∪ B (cid:48) is at most st . Therefore, for every B ∈ J , there is a vertex u B ∈ ∂A contained in ablock at (cid:126)σ -distance ≤ st from B .Let us consider all pairs ( B, u ), where B ∈ J and u ∈ ∂A is a vertex in ablock at (cid:126)σ -distance ≤ st from B and such that there exists a path in A froma vertex a ∈ B to a neighbor of u of length at most 2 st −
1. By the above,each B ∈ J participates in at least one such pair ( B, u B ). A vertex u canbe the second coordinate in at most 2 st + 1 such pairs, since the (cid:126)σ -distanceof u B from B is at most st . Let T be a maximal subset of J such that thecorresponding vertices u B for B ∈ T are pairwise different. Then the set S = { u B | B ∈ T } ⊆ ∂A satisfies | S | = | T | ≤ k . Further, let J = { B ∈ (cid:126)σ : B is at (cid:126)σ -distance ≤ st from some block B (cid:48) with B (cid:48) ∩ S (cid:54) = ∅} . Then J ⊆ J and | J | ≤ | S | (2 st + 1) ≤ k (2 st + 1). Note further, that any interval ofblocks disjoint from J must either all be contained in A or all be disjointfrom A . Next, modify the set J to form J by adding every block B withthe property that the maximal interval of (cid:126)σ \ J containing B has length ≤ t . Since there are at most | T | ≤ k such maximal intervals, we have | J | ≤ (2 st + 1) k + tk and every maximal interval disjoint from J haslength ≥ t + 1. Finally, modify the set J to form J by adding every block B with B ⊆ A . It follows from the assumption that G [ A ∪ ∂A ] is connectedthat J is an interval of (cid:126)σ . Furthermore, setting Q = ∪ B ∈ J B we have A ⊆ Q and | Q \ A | ≤ s | J | ≤ (2 s t + s ) k + stk ≤ s t k + 2 stk as desired. (cid:3) Lemma 5.2.
Let G be a connected vertex transitive graph and let A ⊂ V ( G ) be finite. Then we have (i) If G [ A ∪ ∂A ] is connected, then d iam ( A ) < | ∂A | · (2 d epth ( A ) + 1) . (ii) Let d ≥ , k ≥ , (cid:96) ≥ , m ≥ be integers and assume that | ∂A | ≤ k and d iam ( G ) ≥ mk (2 d + 1) + (cid:96) − d + 1 . If | A | ≥ | V ( G ) |− (cid:96)m , thend epth ( A ) ≥ d + 1 .Proof. Let x, y ∈ A . Let x x . . . x t be a shortest path in G [ A ∪ ∂A ] from x = x to y = x t . For each i , there is a vertex z ∈ ∂A at distance at most d = (cid:100) epth ( A ) from x i . By the minimality of t , each z ∈ ∂A appears in thisway for at most 2 d + 1 indices i ∈ { , , . . . , t } . Thus t < | ∂A | (2 d + 1) andthe same bound holds for the diameter of A .For part (ii), let x , x r ( r = mk (2 d + 1) + (cid:96) − d + 1) be vertices at distance r in G , and let x , x , x , . . . , x r be a shortest path joining them. Considerthe ball B of radius (cid:96) centered at x and balls B i of radius d centered at x (cid:96) +1 − d + i (2 d +1) , for 1 ≤ i ≤ mk . Then B , B , . . . , B mk are pairwise disjoint.Since | B | ≥ (cid:96) + 1, we conclude that every ball of radius d in G contains EPARATIONS IN SYMMETRIC GRAPHS 23 less than km ( | V ( G ) | − (cid:96) ) vertices. If d epth ( A ) ≤ d , then balls of radius d centered at ≤ k vertices in ∂A would cover A , so | A | < | V ( G ) |− (cid:96)m , contraryto our assumption. (cid:3) Lemma 5.3.
Let G be a connected vertex transitive graph, let A ⊆ V ( G ) befinite and set k = | ∂A | . If d iam ( G ) ≥ k + 1) and d epth ( A ) ≥ k + 1 andd epth ( V ( G ) \ ( A ∪ ∂A )) ≥ k +1 , then there exist integers s, t with st ≤ k anda cyclic system (cid:126)σ so that G is ( s, t ) -ring-like and st -cohesive with respectto (cid:126)σ .Proof. We may assume that A is a set which satisfies the assumptions of thelemma, and further, is chosen so that(i) | ∂A | is minimum,(ii) | A | is minimum subject to (i).Note that (ii) implies that G [ A ] is connected and that | A | ≤ | V ( G ) \ ( A ∪ ∂A ) | . We proceed with two claims. (1) d epth ( A ) = k + 1.Suppose (for a contradiction) that d epth ( A ) > k + 1. Choose a vertex x ∈ A at depth k + 2, its neighbor y ∈ A at depth k + 1, and an auto-morphism φ with φ ( x ) = y . Set A = A and A = φ ( A ) and let the sets P, Q, S, T, U, W, X, Y, Z be as given in Figure 1. If | S | , | X | ≥ ( | V ( G ) | − k )then it follows from part (ii) of Lemma 5.2 (with m = 4, d = k , and (cid:96) = (cid:100) k (cid:101) )that d epth ( S ) , d epth ( X ) ≥ k + 1 and by uncrossing that | ∂S | + | ∂X | ≤ k .But then either S or X contradicts our choice of A . Thus, we may as-sume that either | S | or | X | is less than ( | V ( G ) | − k ). If G is finite and | S | ≤ ( | V ( G ) | − k ), then | S ∪ Z | + k ≥ | S ∪ W ∪ Z | ≥ ( | V ( G ) | − k ) so | Z | ≥ ( | V ( G ) |− k ). The same conclusion holds under the assumption that | X | ≤ ( | V ( G ) | − k ), so by Lemma 5.2(ii) we conclude that Z has depth ≥ k + 1. Of course, Z has depth ≥ k + 1 also when G is infinite. It followsfrom our construction that P has depth ≥ k + 1. Now, by uncrossing, either | ∂ ( P ∪ Q ∪ S ∪ T ∪ X ) | < k and this set contradicts the choice of A for (i),or | ∂P | ≤ k and this set contradicts the choice of A for (i) or (ii). (2) For every x ∈ V ( G ) and every n , k ≤ n ≤ d iam ( G ) − k − k −
1, thereexists a vertex set D with | ∂D | = k and B ( x, n ) ⊆ D ⊆ B ( x, n +2 k +2 k − n = k : by claim (1) we see that B ( x, k ) ⊆ φ ( A ), where φ is an automorphism of G mapping a vertex at depth k + 1 in A onto x . Further, part (i) of Lemma 5.2 shows that d iam ( A ) = d iam ( φ ( A )) < k (2 d epth ( A ) + 1) = 2 k + 3 k . Thus, φ ( A ) ⊆ B ( x, k + 2 k − D = φ ( A ) when n = k .Suppose (for a contradiction) that such a set does not exist for n and x ,and choose a maximal set C ⊆ V ( G ) with | ∂C | = k and B ( x, k ) ⊆ C ⊆ B ( x, n + 2 k + 2 k −
1) (such a set exists as shown above). Choose a shortestpath P from x to ∂C , let y be the end of P in ∂A , and choose a vertex z on P at distance k from y . Since B ( x, n ) (cid:54)⊆ C , we have that d ist ( x, z ) < n − k .Using the fact that (2) holds when n = k , choose a set D ⊆ V ( G ) with | ∂D | = k so that B ( z, k ) ⊆ D ⊆ B ( z, k + 3 k − V ( G ) \ ( C ∪ D )contains a ball of radius ≥ k + 1 so d epth ( V ( G ) \ ( C ∪ ∂C ∪ D ∪ ∂D ) ≥ k + 1.If | ∂ ( C ∩ D ) | < k then B ( y, k − ⊆ C ∩ D by construction so C ∩ D hasdepth ≥ k and contradicts our choice of A for (i). Otherwise, it follows fromuncrossing (Lemma 2.1) that | ∂ ( C ∪ D ) | ≤ k . Since d ist ( x, z ) < n − k , wehave C ∪ D ⊆ B ( x, n + 2 k + 2 k − y ∈ D \ C , we have | C ∪ D | > | C | ,so this contradicts our choice of C . This completes the proof of claim (2).Set q = 12 k + 15 k + 3 and h = 6 k + 10 k . Let v − q , v − q +1 , . . . , v q + k + h +2 be the vertex sequence of a shortest path in G . Next, apply (2) to choose sets C, D − , D + ⊆ V ( G ) with | ∂C | = | ∂D − | = | ∂D + | = k so that the followinghold: B ( v , q − k − k ) ⊆ C ⊆ B ( v , q + k + k − B ( v − q , k + 3 k ) ⊆ D − ⊆ B ( v − q , k + 5 k − B ( v q , k + 3 k ) ⊆ D + ⊆ B ( v q , k + 5 k − B ( v q − k − k , k ) ⊆ C ∩ D + and B ( v q + k +2 k , k ) ⊆ D + \ C so these setshave depth ≥ k + 1. Furthermore, B ( v , k ) ⊆ C \ D + and B ( v q + k + h , k ) ⊆ V ( G ) \ ( C ∪ D + ) so these sets have depth ≥ k + 1. It now follows fromour choice of A and uncrossing, that | ∂ ( C ∩ D + ) | = | ∂ ( C \ D + ) | = | ∂ ( D + \ C ) | = | ∂ ( C ∪ D + ) | = k . But then by part (iii) of Lemma 2.1 we find that R = ( D + ∪ ∂D + ) ∩ ∂C satisfies | R | ≥ k . By a similar argument, we findthat L = ( D − ∪ ∂D − ) ∩ ∂C satisfies | L | ≥ k . Thus { L, R } is a partition of ∂C . If x, y ∈ R , then x, y ∈ B ( v q , k + 5 k ) so d ist ( x, y ) ≤ k + 10 k = h .Similarly if x, y ∈ L , then d ist ( x, y ) ≤ h . If x ∈ L and y ∈ R , then d ist ( x, y ) ≥ d ist ( v − q , v q ) − d ist ( v − q , x ) − d ist ( y, v q ) ≥ q − k − k =3 h + 6. Thus C is a (3 h + 6 , h )-tube with boundary partition { L, R } and B ( v q + k + h +2 , h + 2) is disjoint from C so d epth ( V ( G ) \ ( C ∪ ∂C )) ≥ h + 2.Applying the tube lemma to this yields the desired conclusion. (cid:3) We are prepared to complete the proof of our main theorem.
Proof of Theorem 1.9.
Let A ⊆ V ( G ) satisfy the assumptions of the theo-rem. Let A (cid:48) = V ( G ) \ ( A ∪ ∂A ). Observe that ∂A (cid:48) = ∂A . If G is finite, thenby assumption | V ( G ) \ ( A ∪ ∂A ) | ≥ ( | V ( G ) | − k ). By using statement (ii)of Lemma 5.2 applied to the set A (cid:48) with m = 2 and l = d = k , we concludethat d epth ( A (cid:48) ) ≥ k + 1. The same conclusion holds trivially if G is infinite.If d epth ( A ) ≤ k then by (i) of Lemma 5.2 and Theorem 1.11 we find that | A | ≤ k + k so case (i) holds. Note that the parenthetical comment in(i) is a direct application of Theorem 1.5. Otherwise it follows from Lemma5.1 and Lemma 5.3 that (ii) holds. (cid:3) EPARATIONS IN SYMMETRIC GRAPHS 25 Proofs of main corollaries
It remains to prove the main corollaries of Theorem 1.9.
Proof of Theorem 1.3.
Let C = B \ { e } , where e is the identity element ofthe group. Let G be the Cayley graph of G with respect to the symmetricgenerating set C . Observe that ∂A = BA \ A and that A ⊆ BA . We aregoing to apply Theorem 1.9 to the set A ⊂ V ( G ). Let us first assume that G [ A ∪ ∂A ] is connected. Since | BA | < | A | + | A | / , and since A ⊆ BA , wesee that k = | ∂A | < | A | / . Since G is infinite, the assumptions of Theorem1.9 are satisfied and we have one of the outcomes (i) or (ii) of Theorem 1.9. If(ii) holds, then let B n ( n ∈ Z ) be the blocks of imprimitivity correspondingto the ( s, t )-ring-like structure of G , where B contains the identity elementof the group. Let N = { g ∈ G | B n g = B n for every n ∈ Z } . Since G acts regularly on the vertex set of its Cayley graph G by rightmultiplication, we conclude that | N | ≤ | B | = s ≤ k . It is easy to see that N is a normal subgroup of G and that G /N , which is acting transitively onthe two-way-infinite quotient, is either cyclic or dihedral. This gives one ofthe outcomes of the theorem.Suppose now that we have outcome (i) of Theorem 1.9. In that casewe conclude, in particular, that | A | ≤ k + k . If we use the fact that k < | A | / , we conclude that | A | < | A | + | A | / ≤ | A | . This contradiction completes the proof when G [ A ∪ ∂A ] is connected.If G [ A ∪ ∂A ] is not connected, then A can be partitioned into sets A , . . . , A l such that the sets A i ∪ ∂A i (1 ≤ i ≤ l ) induce connected subgraphs of G and partition A ∪ ∂A . Let us define k i = | ∂A i | (1 ≤ i ≤ l ). We may assumethat we have the first outcome of Theorem 1.9 for each A i , which, as shownabove, implies that k i ≥ | A i | / . It follows that k = l (cid:88) i =1 k i ≥ l (cid:88) i =1 | A i | / ≥ (cid:18) l (cid:88) i =1 | A i | (cid:19) / = 12 | A | / . (cid:3) Next we turn towards the proof of Corollary 1.19. In this corollary we usethe notion of the tree-width. This parameter, which has been introducedin the graph minors theory, formalizes the notion of a graph being “tree-like”. If G is a graph and T is a tree, then a tree-decomposition of G in T is family of subtrees T v ⊆ T ( v ∈ V ( G )), such that whenever uv ∈ E ( G ),the corresponding subtrees intersect, T v ∩ T u (cid:54) = ∅ . The order of the tree-decomposition is defined as the maximum cardinality of the sets Y t = { v ∈ V ( G ) | t ∈ V ( T v ) } taken over all vertices t ∈ V ( T ). Finally, the tree-width of G is the minimum order of a tree-decomposition of G minus 1. One can only consider tree-decompositions without “redundancies” in which case theset Y s ∩ Y t is a separator of G for every edge st ∈ E ( T ). This explains whythe notion of the tree-width is related to the subject of this paper.We will make use of the following result; see [30] or [13, Section 11.2]. Weinclude a short proof for completeness. Lemma 6.1 (Balanced Separator Lemma) . If a graph G has tree-width lessthan k , then for every vertex set W ⊆ V ( G ) there exists a set S ⊆ V ( G ) with | S | ≤ k such that every connected component of G − S contains at most | W | vertices from W .Proof. Let us consider a tree-decomposition of G in a tree T of order at most k , and assume that subject to these conditions, T is minimum possible. Let st ∈ E ( T ) be an edge of T . If Y t ⊆ Y s , then we could replace the tree T bythe smaller tree T (cid:48) = T /st obtained by contracting the edge st , and replaceeach T v containing the edge st by the subtree T (cid:48) v = T v /st ⊆ T (cid:48) to obtaina tree-decomposition of the same order and having smaller tree. It followsthat the set Y t ∩ Y s separates the graph G , i.e. G − ( Y t ∩ Y s ) is disconnected.Let us observe that this also implies that for each leaf t in T , there is avertex v ∈ V ( G ) such that T v = { t } .Suppose that st ∈ E ( T ) and that there is a component C of G − ( Y t ∩ Y s )containing more than | W | vertices from W . Let T and T be the subtreesof T − st containing the vertex s and t , respectively. If V ( C ) ⊆ ∪ p ∈ V ( T ) Y p ,then we orient the edge st from t to s , otherwise from s to t . By repeatingthis for all edges of T , we get a (partial) orientation of the edges of T withthe property that the out-degree of each vertex is at most 1. Therefore,there is a vertex t ∈ V ( T ) whose out-degree is 0. Clearly, the set S = Y t satisfies the conclusion of the lemma. (cid:3) Proof of Corollary 1.19.
Let G be a connected finite vertex transitive graphand suppose that the tree-width of G is less than k . Let us consider a tree-decomposition of G in a tree T of order at most k , and assume that subjectto these conditions, T is minimum possible. Let t ∈ V ( T ) be a leaf of T .As shown in the proof of the Balanced Separator Lemma, there is a vertex v ∈ V ( G ) such that T v = { t } . All neighbors of this vertex are in Y t and | Y t | ≤ k , hence the degree of v in G is at most k −
1. If the diameter of G isless than 31( k + 1) , then we have the last outcome of the corollary, so wemay assume henceforth that the diameter is at least 31( k + 1) .Let W be a path in G joining two vertices that are at distance 31( k + 1) .By the Balanced Separator Lemma 6.1, there exists a set S ⊂ V ( G ) with | S | ≤ k such that no component of G − S contains more than | W | verticesof W . For each vertex u ∈ S , let A u be the vertex set consisting of allvertices in W whose distance from u in G is at most k . Let W u be thesmallest segment on W that contains A u . As any two vertices in A u are atdistance at most 2 k from each other, we have | W u | ≤ k + 1. Since | W \ ( ∪ u ∈ S W u ) | ≥ | W | − k (2 k + 1) > | W | EPARATIONS IN SYMMETRIC GRAPHS 27 there are at least two components of G − S that contain a vertex in W \ ( ∪ u ∈ S W u ). Let A be the vertex set of the smaller one of these two com-ponents. Then | A | < | V ( G ) | and ∂A ⊆ S , thus | ∂A | ≤ k . Also, G [ A ] isconnected and since A contains a vertex in W \ ( ∪ u ∈ S W u ), we have that d epth ( A ) > k . By Theorem 1.9 we conclude that G is ( s, t )-ring-like with2 st ≤ k , i.e., we have the outcome (i) of Corollary 1.19. This completes theproof. (cid:3) References [1] L. Babai, Local expansion of vertex-transitive graphs and random generation in finitegroups, in
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