Crown-free highly arc-transitive digraphs

Transcription

1 Crown-free highly arc-transitive digraphs Daniela Amato and John K Truss University of Leeds 1. Abstract We construct a family of infinite, non-locally finite highly arc-transitive digraphs which do not have universal reachability relation and which omit special digraphs called crowns. Moreover, there is no homomorphism from any of our digraphs onto Z. The methods are adapted from [5] and [6]. 1 Introduction The background to the work presented here is the paper of Cameron, Praeger, and Wormald [1] which gives some general theory about the structure of highly arc-transitive digraphs (those whose automorphism group acts transitively on the family of s-arcs for all s), and gives several examples and questions about them. A key result introduces a certain relation on edges, called the reachability relation (described below), and establishes a link between the existence of a homomorphism from the digraph onto the natural digraph on Z (in which case we say that it has property Z ) and the reachability relation assuming just 1-arc transitivity. Precisely, they show the following: if D is a connected 1-arc transitive digraph and the reachability relation for D is universal, then D does not have property Z. Arising from this result, they raised the problem of constructing highly arc-transitive digraphs without property Z. In response to this, examples were given in [9] of locally finite, highly arctransitive digraphs without property Z. The reachability relation for these digraphs is however not universal, and another question posed in [1] was whether there is a locally finite highly arc-transitive digraph for which the reachability relation is universal. This question remains open. Meanwhile, also responding to a question in [1], Evans [5] gave a quite different method for constructing a highly arc-transitive digraph that does not have property Z, based on a modification of Fraïssé amalgamation. The resulting structure is no longer locally finite; however it turns out, as we remark below, that the reachability relation is universal. The examples we construct are based on those in [5] and [6]. In the non-locally finite case, no examples were known to us of highly arc-transitive digraphs without property Z for which the reachability relation is not universal, so one of our goals here is to construct a digraph with these properties. We also demonstrate the flexibility of Evans method by applying it to modified situations in which certain substructures are omitted, namely ones termed crowns, which are alternating walks forming a circuit. The paper is organized as follows. In Section 2 we give some basic definitions and background on the reachability relation, property Z and ends of graphs. In Section 3 we construct an infinite, non-locally finite highly arc-transitive digraph without property Z for which the reachability relation is not universal. Finally, in Section 4 we give the proof of the universality of the reachability relation for the digraphs constructed in [5] and [6]. 1 This work was supported by a grant from the EPSRC. Mathematics Subject Classification: 05C20, 05C38 1

2 2 Preliminaries Our digraphs will have no loops or multiple edges and the vertex and edge sets of a digraph D are written V D and ED respectively. The out-valency (in-valency) of a vertex α is the size of the set {u V D (α, u) ED}; ({u V D (u, α) ED} respectively). A digraph is locally finite if all in- and out-valencies are finite. If the automorphism group Aut(D) of D is transitive on V D, all vertices in D have the same in-valency and all vertices have same out-valency. An s-arc for s 0 in D is a sequence u 0 u 1... u s of s + 1 vertices such that (u i, u i+1 ) ED and u i 1 u i+1 for i {0,..., s 1}. We say that D is s-arc transitive if Aut(D) is transitive on the set of s-arcs, and it is highly arc-transitive if it is s-arc transitive for all s Property Z and the reachability relation We write Z for the digraph on Z whose only edges are (i, i + 1) for i Z. A digraph D has property Z if there is a homomorphism from D onto Z, and this is equivalent to saying that there are no unbalanced cycles in D, where by an unbalanced cycle we mean a cycle with a unequal number of forward and backward edges. An alternating walk in a digraph D is a sequence (x 0, x 1,..., x n ) of vertices of D such that either (x 2i 1, x 2i ) and (x 2i+1, x 2i ) are edges for all i, or (x 2i, x 2i 1 ) and (x 2i, x 2i+1 ) are edges for all i. If e and e are edges in D and there is an alternating walk (x 0, x 1,..., x n ) such that (x 0, x 1 ) is e and either (x n 1, x n ) or (x n, x n 1 ) is e, then e is said to be reachable from e by an alternating walk. This is denoted by eae. Clearly A is an equivalence relation on ED. The equivalence class containing e will be denoted by A(e). If D is 1-arc-transitive, the digraphs A(e), e V D, induced on A(e), are all isomorphic to a fixed digraph, which is denoted by (D). Proposition 2.1 [1] If D is a connected 1-arc transitive digraph, then (D) is 1-arc transitive and connected. Furthermore, either (a) A is the universal relation on ED and (D) = D, or (b) (D) is bipartite. In the proof it is shown that for A to be universal it suffices that A(e) contain a 2-arc, otherwise (D) is bipartite. Lemma 2.2 [1] Let D be a connected 1-arc transitive digraph. If the reachability relation A for D is universal then D does not have property Z. It was shown in [12] that highly arc-transitive digraphs with finite and unequal in- and outvalency always have property Z. Hence the candidates for highly arc-transitive digraphs without property Z and which are locally finite, have equal in- and out-valencies. As mentioned in the introduction, the digraphs constructed in [9] are examples of highly arc-transitive digraphs without property Z. For each n 3, the authors construct a digraph D with both in-valency and out-valency equal to 2, and for which the bipartite digraph (D) is a 2n-crown (see Section 3 for the definition). The digraphs constructed in [5] and [6] are non-locally finite examples of highly arc-transitive digraphs without property Z. In Section 4 we show that the reachability relation in these digraphs is universal. 2.2 Descendant sets For a vertex a in D the descendant set desc(a) of a is the subdigraph induced on the set of all vertices which can be reached by a directed path (that is, a s-arc for some s) from a. If D is transitive, any two descendant sets in D are isomorphic. A subdigraph A of D is finitely generated if it is the union of finitely many descendant sets. For X V D, we let desc(x) = x X desc(x). Also, for X, Y V D, we let desc(x, Y ) := desc(x) desc(y ). 2

3 In [1] it is shown that an infinite, connected highly arc-transitive digraph D contains no finite directed cycle. So the subdigraph Γ := desc(α) is rooted with root α in the sense that within Γ the vertex α has in-valency 0 and every vertex can be reached by a directed path from α. Also, a result in [1] (see Proposition 3.10) shows that if x, y V D and there are an s-arc and a t-arc from x to y, then s = t. So Γ is a layered subdigraph of D, with layers the sets Γ s (α) consisting of the set of points which can be reached by an s-arc from α. If, moreover, the automorphism group of D is primitive on V D, it is shown in [2] that Γ is tree-like. The highly arc-transitive digraphs in [5], [6] and [9] have the property that the descendant set of a vertex is a q-ary rooted tree for some q 2. The digraphs we construct in this work will also have this property. 2.3 Ends of graphs Let X be an infinite connected graph. By a ray in X we mean an infinite sequence {v i } i N of distinct vertices such that v i is adjacent to v i+1 for all i N. The ends of the graph X are the equivalence classes of rays where R 1 and R 2 are said to be equivalent if there is a third ray R 3 such that R 3 contains infinitely many vertices from both R 1 and R 2 (see [10]). There are several ways to rephrase this definition. One is to say that R 1 and R 2 are in the same end if and only if for every finite set F V X there is a path in X\F connecting a vertex in R 1 to a vertex in R 2. Or we could say that R 1 and R 2 are not in the same end if and only if one can find a finite subset F of vertices and distinct components C 1 and C 2 of X\F such that C 1 contains infinitely many vertices of R 1 and C 2 contains infinitely many vertices of R 2. Of course, a connected, infinite locally finite graph has at least one end. It is known that a vertex-transitive graph has 1,2 or 2 ℵ0 ends. Roughly speaking, the number of ends of a undirected graph is the number of ways of going to infinity. In particular, a locally finite connected graph X has more than one end if one can find a finite subset F of vertices such that X\F has more than one infinite component. And X has infinitely many ends if for all N N there is a finite set F of vertices of X such that X\F has more than N infinite components. The ends of a digraph D are the ends of the underlying graph of D. A number of results about ends of highly arc-transitive digraphs are known. The digraph Z is a highly arc-transitive digraph with two ends, while an infinite, directed regular tree is highly arc-transitive and has infinitely many ends ( where by regular we mean that every vertex has the same in-valency, and the same out-valency). The digraph in [9] also has infinitely many ends. On the other hand, the digraphs constructed in [5] and [6] have only one end (see Section 4). 3 Crown-free highly arc-transitive digraphs In [5] and [6] Evans uses a version of Fraïssé-amalgamation (see [4] for a survey). This method is widely used throughout model theory to construct structures with a large supply of automorphisms. Roughly speaking, we construct a digraph by gluing together smaller digraphs (from a certain class of digraphs) in a systematic way so that isomorphisms between finitely generated subdigraphs extend to an automorphism of the large digraph. Let T be a q-ary tree for some fixed q > 1. Throughout we work with digraphs for which the descendant set of any vertex is isomorphic to T. So a finitely generated digraph is a finite union of copies of T. A set X of vertices of a digraph A is independent if the intersection desc(x, y) is empty for all x y in X, and X is descendant-closed in A, denoted by X A, if desc(x) X for all x X. For a, b A, by an alternating walk from a to b we mean an alternating walk (a 0,..., a k ) with a 0 = a and a k = b. In particular, when k = 2n and a 2n = a 0, with a 1 a 0, the circuit (a 0,..., a 2n ) is called a 2n-crown. In this section we construct an infinite, connected highly arc-transitive digraph D which is crownfree, that is, D does not embed a 2n-crown, for any n 2. We start with a class C whose members are digraphs A which are a union of finitely many copies of T. Also, for a in V A and vertices x, y in desc(a), there will be no alternating walk from x to y containing vertices from A\ desc(a) (we say 3

4 desc(a) is -embedded in A). We shall prove that this condition is preserved when we amalgamate two members of C. The first step towards the construction of the crown-free digraph D is the construction, for each m 2, of a highly arc-transitive digraph D(m) which does not embed any 2(n + 1)-crown, for n {1,..., m 1}. It will be clear from the construction that D(m) properly embeds into D(m 1). 3.1 The digraphs D(m) Definition 3.1 Let X A. We write X m A to mean that for x, y X and n {1,..., m 1}, there is no alternating walk from x to y of length 2n and which intersects A\X. The definition says that a descendant-closed subdigraph X of A is m -embedded in A if, for n {1,..., m 1}, any alternating walk of length 2n from x X to y X must lie entirely in X The class C m of digraphs Definition 3.2 Let C m be the class of digraphs A satisfying the following conditions: C1: for all a A, desc(a) is isomorphic to T, C2: A is finitely generated, C3: for all a, b A the intersection desc(a, b) is finitely generated, C4: desc(a) m A for all a A. Let n {1,..., m 1}. It follows from Condition C4 that a digraph A in C m does not embed a 2(n + 1)-crown. For if a 0,..., a 2n+2 is a 2(n + 1)-crown in A, (a 2,..., a 2n+2 ) is an alternating walk of length 2n from a 2 desc(a 1 ) to a 2n+2 = a 0 desc(a 1 ). But this means desc(a 1 ) is not m -embedded in A. The following is as in [7] (Lemma 2.4). Lemma 3.3 Let C C m and let A, B be finitely generated subdigraphs of C, with A, B C. Then the intersection A B is finitely generated. Proof. Let {a 1,..., a n } and {b 1,..., b s } be generating sets in C for A and B respectively. Then A B = desc(a i, b j ). i {1,...,n},j {1,...,s} Since A, B C and C C m, the intersection desc(a i, b j ) is finitely generated for all i, j. Therefore A B is finitely generated. Lemma 3.4 Let X m Y and Y m Z. Then X m Z. Proof. Let n {1,..., m 1} and let (a 0, a 1,..., a 2n ) be an alternating walk with a 0 = x, a 2n = x where x, x X. Then (a 0, a 1,..., a 2n ) is an alternating walk in Z from a 0 Y to a 2n Y, so as Y m Z, it does not intersect Z\Y. Hence it is an alternating walk in Y from a 0 X to a 2n X, so as X m Y, it does not intersect Y \X, and therefore lies entirely in X. Hence X m Z. Lemma 3.5 (Hereditary property) If A C m and X m A is a finitely generated subdigraph of A, then X C m. Proof. Clearly X satisfies C1 as it is descendant closed. Also, X satisfies C2 by assumption, and C3 follows from the same property for A. For C4, let x X. Since A C m and x A, desc(x) m A. So for any u, v in desc(x) and n {1,..., m 1}, an alternating walk from u to v of length 2n must lie entirely in desc(x). Hence desc(x) m X. Thus X C m. 4

5 3.1.2 Amalgamation We now define our notion of amalgamation for digraphs in C m. Let A, B C m. Let U be a finite subset of vertices of A which is independent and such that the subdigraph X on desc(u) is m -embedded in A. Similarly let V be a finite and independent subset of B with Y := desc(v ) m -embedded in B, where U = V = n. So X, Y C m. Now, by independence, X and Y are isomorphic to the disjoint union of n copies of T, therefore there is an isomorphism from X to Y. We use the isomorphism to identify X and Y, and define the free amalgam (A, X) (B, Y ) to be the disjoint union of A and B over X and Y (that is, the digraph with vertex set the union of the vertices of A and B with X and Y identified, and edge set the union of the edge sets of A and B). We show that the amalgam C := (A, X) (B, Y ) lies in C m, and A, B m C. It is clear that C satisfies C1 and, since A and B are finitely generated, C is too. For C3, let a, b C. If a, b A or a, b B, then desc(a, b) is finitely generated as A and B are descendant closed in C. Suppose a A\B and b B\A. Since the intersection desc(a, b) is contained in A B and A B = X, desc(a, b) = desc(a, X) desc(b, X). Now, X m A and therefore X C m. It follows by Lemma 3.3 that both desc(a, X) and desc(b, X) are finitely generated subdigraphs of X which are -embedded in X, so by Lemma 3.3 again, desc(a, X) desc(b, X) is finitely generated. But this equals desc(a, b), and so this is also finitely generated. We now show that A is m -embedded in C. We must prove that any alternating walk of length 2n connecting two vertices in A lies entirely in A (so no new alternating walk is created by the amalgamation). Let n {1,..., m 1} and let (a 0, a 1,..., a 2n ) be an alternating walk from a 0 A to a 2n A. Suppose there is j {1,..., 2n 1} such that a j B\A. Then (a 0,..., a j ) is an alternating walk from a vertex a 0 in A to a vertex a j in B\A. Since A B = X, there is r {1,..., j 1} such that a r X. Similarly, there is s {j + 1,.., 2n} with a s X. So (a r, a r+1,..., a s 1, a s ) is an alternating walk of length at most 2n from a r X to a s X and which contains the vertex a j B\X. This contradicts the assumption that X is m -embedded in B. Hence A m C, and similarly B is m -embedded in C. Finally, for C4 let c C. Without loss of generality, let c A. Since desc(c) m A and A m C, it follows by Lemma 3.4 that desc(c) m C. Hence C m has the amalgamation property The construction The next lemma follows from the similar result in [6]. Lemma 3.6 (a) There are countably many isomorphism types of digraphs in C m. (b) Let A, B C m. Then the set of m -embeddings of A into B is countable. From now on we refer to a m -embedding simply as an embedding. We are now ready to state our main result. Theorem 3.7 There is a countable digraph D := D(m) satisfying conditions C1, C3 and C4 and with the following properties: (a) ( m -extension) for all A, B C m with A m D and embeddings φ : A B, there is an embedding ψ : B D with ψ φ = id, (b) ( m -homogeneity) every isomorphism f : A B, with A, B m D and A, B C m, extends to an automorphism of D. Moreover, D is unique (up to isomorphism) with these properties. The construction of the digraph D(m) is very similar to the constructions in [3], [5] and [6]. We give a sketch of the proof. We start with any member D 0 of C and construct a sequence D 0 m D 1 m m D i m, (1) 5

6 of elements D i in C m, where D i+1 is built up inductively from D i using the amalgamation property for m -embedded subdigraphs. By Lemma 3.6, there are only countably many amalgamations to perform. We then let D(m) := i<ω D i. Clearly D(m) is countable and satisfies conditions C1, C3 and C4. The extension property (a) for m -embeddings follows from the construction, and to show that D(m) is m -homogeneous, one uses a back-and-forth argument. Theorem 3.8 The digraph D(m) is connected. Proof. We write D(m) as D for short. We show that for x, y in V D, there is a path from x to y in the underlying graph of D. If x desc(y) or y desc(x), or if desc(x) and desc(y) intersect, then such a path exists. Now suppose that desc(x) and desc(y) do not intersect, and let X be the subdigraph of D on desc(x) desc(y). We have two cases to consider. Case 1: X is m -embedded in D. Since X is finitely generated, X m D i for some digraph D i in sequence (1) above. Let T be a q-ary tree with root a. There is a m -embedded subdigraph Y of T which is isomorphic to X, since desc(x) and desc(y) are disjoint, and we can take incomparable vertices of T at distance greater than 2 corresponding to x and y. Amalgamate D i and T over X and Y. By the m -extension property, there is u in V D (namely the image of the root of T ) such that X is a subdigraph of desc(u). Therefore there is a path from x to y in D. Case 2: X is not m -embedded in D. In this case, there are vertices u and v in X and an alternating walk of length 2n from u to v intersecting D\X, for some n {1,..., m 1}. By condition C4, both desc(x) and desc(y) are m -embedded in D, so u and v cannot both lie in the same one of desc(x), desc(y), and we must have u desc(x) and v desc(y) or vice versa. So there is a path from x to y in D. Moreover, Theorem 3.9 The digraph D(m): (i) properly embeds in D(m 1), (if m 3), (ii) is 2(n + 1)-crown-free, for all n {1,..., m 1}, (iii) is highly arc-transitive, (iv) does not have property Z, and (v) has universal reachability relation. Proof. (i) That D(m) embeds in D(m 1) follows from the m 1 -extension property, C m C m 1, and the evident fact that for any X, Y, X m Y X m 1 Y. The embedding is proper because D(m 1) embeds a 2m-crown (as there is an element of C m 1 embedding a 2m-crown) but D(m) does not. (ii) follows from condition C4, as remarked earlier. Again we write D for D(m). For (iii) we first show that the automorphism group G of D is transitive on vertices. Let x, y V D. By condition C1, desc(x) = desc(y), and by condition C4, desc(x) and desc(y) are m -embedded in D. So by m -homogeneity (Theorem 3.7(b)), any isomorphism ϕ from desc(x) to desc(y) with ϕ(x) = y extends to an automorphism of D. For high arc-transitivity, let γ, γ be s-arcs in D for some s 1. By vertex-transitivity, we may assume that γ and γ have the same initial vertex, x say. Since desc(x) is a q-ary tree, there is an automorphism of desc(x) taking γ toγ, and this extends to an automorphism of D by m -homogeneity. To prove that D does not have property Z, we show that it embeds an unbalanced cycle. Let T be a q-ary tree with root a. Let x desc 1 (a) and let y be a vertex in desc 2 (a) which is not in desc(x). Clearly the subdigraph X of T on desc(x) desc(y) is m -embedded in desc(a). Now let T be another copy of T with root b, and let x and y be the copies of x and y in T. Let Y be the subdigraph of T on desc(x ) desc(y ). Now amalgamate T and T over desc(x) desc(y) and desc(x ) desc(y ), identifying x with y and y with x. The resulting digraph A is an element of 6

7 C m and therefore embeds into D (by the m -extension property). Since A embeds the digraph with vertex set {a, x, v, b, y, u} and edge set {(a, x), (a, u), (u, y), (b, y), (b, v), (v, x)}, it follows that D embeds an unbalanced cycle. We now show that D has universal reachability relation. Since D is connected and 1-arc transitive, by Lemma 2.1 it suffices to show that (D) contains a 2-arc. Let x V D and let (x, y, z) be a 2-arc. Since desc(x) is a q-ary tree, there is y y in desc 1 (x). Let X be the subdigraph of D on desc(z) desc(y ). It is clear that X is m -embedded in desc(x). Now let r > m + 1 and let (a 0,..., a 2r ) be an alternating walk, with a 1 a 0, such that for each i {0,..., r} the subdigraph on desc(a 2i ) is isomorphic to T and desc(a 2i ) does not intersect desc(a 2j ), for all i j. Let A be the digraph on r j=1 desc(a 2j 1). It is easy to verify that A lies in C m 1. Note that the subdigraph Y of A on desc(a 0 ) desc(a 2r ) is m -embedded in A (since (a 0,..., a 2r ) is the unique alternating walk connecting vertices of Y and which intersects A\Y, and this walk has length greater than 2m). Finally, amalgamate desc(x) and A over X and Y, identifying a 0 with y and a 2r with z. By the m -extension property, the edge f = (y, z) is reachable from e = (x, y) by an alternating walk in D. Hence eaf and therefore (D) contains a 2-arc. So for each m 2 we have constructed a digraph which is 2n-crown-free for all n {2,..., m}. A natural question arising here is the following: Question: Is there a highly arc-transitive digraph which is 2m-crown-free but which embeds a 2n-crown for some n {2,..., m}? 3.2 A crown-free digraph By Theorem 3.9 we have a strictly decreasing sequence D(2) D(3) D(m) of digraphs. We want to consider the intersection D( ) = m 2 D(m). It is clear that D( ) is a countable digraph which does not embed a 2n-crown, for any n 2 (so is crown-free). However D( ) is not uniquely determined as it may depend on the particular way in which each D(m + 1) is embedded in D(m) (for instance, it may even be empty, or if non-empty, it is not clear that it is connected). We form the desired D( ) rather as the limit digraph of the class C of digraphs A satisfying conditions C1, C2, C3 and C4 : desc(a) A for all a A, where for a subdigraph X of A, we write X A to mean that X is descendant-closed in A and, for x, y X, there is no alternating walk from x to y intersecting A\X. This implies that A is crownfree. It is easy to verify that the results in the previous section hold with m replaced by. The limit digraph D( ) satisfies conditions C1, C3 and C4 and has the -extension property and is -homogeneous. Also, following similar reasoning to the proof of Theorem 3.8, one can show D( ) is connected. It seems likely that D( ) can then be written as an intersection of the form m 2 D(m), though we have not verified this. Certainly D( ) can be embedded in each D(m). Theorem 3.10 The digraph D( ) has the following properties: (i) it is highly arc-transitive, (ii) it does not have property Z, and (iii) the reachability relation is not universal. Proof. The proof of parts (i) and (ii) are similar to the proof of the corresponding results in Theorem 3.9. It remains to prove (iii). Again writing D instead of D( ), suppose for a contradiction that D has universal reachability relation. Then the class A(e) of an edge e in D contains a 2-arc (x, y, z). Let e = (x, y) and f = (y, z). Since eaf, there is an alternating walk (x 0, x 1,..., x 2n+1 ) with n 2 and x 0 = x 2n+1 = y, x 1 = x 7

8 and x 2n = z. It follows that (x 2, x 3,..., x 2n ) is an alternating walk from the vertex x 2 desc(x) to z desc(x), where x 3 does not lie in desc(x). This contradicts C4. Hence A is not universal. So D( ) is a non-locally finite, highly arc-transitive digraph without property Z and for which the reachability relation is not universal. 4 Highly arc-transitive digraphs with universal reachability relation We conclude by showing that the graphs in [6] have universal reachability relation, and just one end, since the techniques needed are similar to those earlier in the paper (even though these graphs are certainly not crown-free). The digraphs M constructed there are infinite, non-locally-finite, and highly arc-transitive, and their automorphism groups act primitively on V M. They are constructed as the limit graph of a class C of digraphs A satisfying conditions C1, C2 and C3 above, and C4 : desc(a) + A for all a A, where a subdigraph X of A is + -embedded in A if X is descendant-closed and, for u in A, if desc(u)\x is finite then u X. By Lemma 2.2 in [6], + is transitive. As in the previous section, one obtains a countable digraph M satisfying conditions C1, C3 and C4 and having the following properties: (1) ( + -extension ) for all A, B C with A + M and + -embedding φ : A B, there is a + -embedding ψ : B M with ψ φ = id, and (2) ( + -homogeneity) every isomorphism f : A B, with A, B + M and A, B C, extends to an automorphism of M. In this section we show that the digraph M has universal reachability relation, and that M has only one end. Theorem 4.1 The digraph M has universal reachability relation. Proof. By Lemma 2.1 it suffices to show that the subdigraph (M) contains a 2-arc. Let (x, y, z) be a 2-arc in M, and let e = (x, y) and f = (y, z). Let y y be a vertex in desc 1 (x) and let X be the subdigraph of desc(x) on desc(z) desc(y ). It is easy to verify that X is + -embedded in desc(x). Now let T and T be q-ary rooted trees, with roots a 1 and a 3 respectively. Let a 2 desc 1 T (a 1) and let a 2 desc 1 T (a 3). The subdigraph of T on desc(a 2 ) is + -embedded in T. Similarly, desc(a 2) is + -embedded in T. Amalgamate T and T over desc(a 2 ) and desc(a 2), identifying a 2 and a 2. The resulting digraph A is member of C with generating set a 1, a 3, where desc(a 1 ) desc(a 3 ) = desc(a 2 ). Next let a 0 a 2 be a vertex in desc 1 A (a 1), and a 4 a 2 be a vertex in desc 1 A (a 3). Then (a 0, a 1, a 2, a 3, a 4 ) is an alternating walk in A. The subdigraph Y of A on desc(a 0 ) desc(a 4 ) is + -embedded in A, and Y is the union of two disjoint copies of T and therefore isomorphic to X. Finally, amalgamate desc(x) and A over X and Y, identifying a 0 with z and a 4 with y. The resulting digraph B is a member of C, so by the + -extension property, it follows there is an alternating walk in M from e to f. Let S be a finite subset of V M. The closure cl M (S) of the set S in M is defined in [6] as the set cl M (S) := {u V M desc(u)\ desc(x) is finite}. It is clear that desc(s) cl M (S) and cl M (S) is + -embedded in M. Moreover, it is shown in [7] that cl M (S) is finitely generated. Theorem 4.2 The digraph M has only one end. 8

9 Proof. It is enough to show that in the underlying graph of M there are infinitely many disjoint paths between any two vertices in M. Let a, b V M and let A 0 := cl M (a, b). So A 0 is a finitely generated + -embedded subdigraph of M. Now suppose inductively that we have found a sequence A 0 + A A i of finitely generated + -embedded subdigraphs of M. We aim to find a + -embedded subdigraph A i+1 of M with A i + A i+1 and which contains a path γ i from a to b such that γ i A i = {a, b}. To achieve this, we follow three steps: Step 1: Amalgamate A i with a copy of T over the empty set. The resulting digraph is a member of C, so by the extension property, there is a vertex y i in V M such that desc(y i ) A i = and the subdigraph B of M on desc(y i ) A i is + -embedded in M. In particular, the intersections of desc(y i ) with each of desc(a) and desc(b) are empty. Let X and Y be the subdigraphs of B on desc(a) desc(y i ) and desc(b) desc(y i ) respectively. Since each of X and Y is the union of disjoint copies of T, we have X, Y + B. It follows by transitivity of + that X, Y + M. Step 2: Now let T be a new copy of the tree, and let X be a copy of X in T, with X + T. Amalgamate B and T over X and X. Note that B is + -embedded in the amalgam. Then by the extension property, there is a vertex c i (the image of the root of T ) in M such that the subdigraph C of M on B desc(c i ) is + -embedded in M, with B desc(c i ) = desc(a) desc(y i ) and B + C. Note that Y + B + C, so Y + C. Step 3: One can use similar reasoning (as in the previous step) for C and Y to obtain a vertex d i V M such that the subdigraph D on C desc(d i ) is + -embedded in M, where C desc(d i ) = desc(b) desc(y i ) and C + D. We then let A i+1 := D, which is a + -embedded subdigraph of M, with A i + A i+1 and this contains path γ i from a to b with a, c i, y i, d i, b γ i. Since γ i A i = {a, b}, this gives a new path from a to b, and consequently there are infinitely many such paths in all. Using a similar argument, one can show that the digraph constructed in [5] also has only one end. The class D of digraphs A used in the construction this time satisfies only the first three conditions C1, C2 and C3, and the limit digraph is imprimitive. (Condition C4 was added in [6] to guarantee primitivity of the limit digraph; Peter Neumann had asked in [11] if this was possible.) References [1] Peter J. Cameron, Cheryl E. Praeger and Nicholas C. Wormald, Infinite highly arc-transitive digraphs and universal covering digraphs, Combinatorica 13 (1993), no 4, [2] Daniela Amato, Descendants in infinite primitive and highly arc-transitive digraphs, submitted. [3] Daniela Amato and John Truss, Some constructions of highly arc-transitive digraphs, submitted. [4] David M. Evans, Examples of ℵ 0 -categorical structures, in Richard Kaye and Dugald Macpherson (eds.), Automorphisms of first-order structures, O.U.P. Oxford (1994), [5] David M. Evans, An infinite highly arc-transitive digraph, Europ. J. Combin. 18 (1997), no. 3, [6] David M. Evans, Suborbits in infinite primitive permutation groups, Bull. London Math. Soc. 33 (2001), no. 5, [7] Josephine Emms and David M. Evans, Constructing continuum many countable, primitive, unbalanced digraphs, to appear in Discrete Mathematics. [8] R. Gray and R. Möller, Locally-finite connected-homogeneous digraphs, in preparation. 9

10 [9] Aleksander Malnič, Dragan Marušič, Norbert Seifter, Boris Zgrablić, Highly arc-transitive digraphs with no homomorphism onto Z, Combinatorica 22 (2002), no. 3, [10] Rögnvaldur G. Möller, Groups acting on locally finite graphs - a survey of the infinitely ended case, Groups 93 Galway/St Andrews, Vol. 2, London Math. Soc. Lecture Notes Series 212, Cambridge University Press, Cambridge 1995, pp [11] Peter M. Neumann, Postcript to review of [1], Bull. London Math. Soc. 24 (1992), [12] Cheryl E. Praeger, On homomorphic images of edge-transitive directed graphs, Australas. J. Combin. 3 (1991),