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An infinite family of sharply two-arc transitive digraphs
Electronic Notes in Discrete Mathematics 29 (2007) 243–247 www.elsevier.com/locate/endm
An infinite family of sharply two-arc transitive digraphs S`onia P. Mansilla 1,2 Departament de Matem` atica Aplicada IV Universitat Polit`ecnica de Catalunya Av. del Canal Ol´ımpic, s/n, 08860 Castelldefels, Spain
Abstract We disclaim Conjecture 1 posed by Seifter in [N. Seifter, Transitive digraphs with more than one end, Discrete Math., to appear], that stated that a connected locally finite digraph with more than one end is either 0-, 1- or highly arc transitive. We describe in this work an infinite family of 2-arc transitive two-ended digraphs, that are not 3-arc transitive. Keywords: Highly arc transitive, k-arc transitive digraphs, ends of graphs.
1
Introduction
Since decades transitive graphs have been the subject of much investigation. The study of k-edge transitive (undirected) graphs goes back to Tutte [7], who showed that finite cubic graphs cannot be k-edge transitive for k > 5. Weiss [8] proved several years later that the only finite connected k-edge transitive 1
Work supported partly by the Spanish Research Council (Comisi´ on Interministerial de Ciencia y Tecnolog´ıa) under the project MTM2005-08990-C02-01. 2 Email: [email protected] 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.041
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S.P. Mansilla / Electronic Notes in Discrete Mathematics 29 (2007) 243–247
graphs with k ≥ 8 are the cycles. In 1991 this result was extended to infinite graphs of polynomial growth: Seifter in [5] proved that a connected, vertextransitive graph of polynomial growth with vertex degrees at least 3 cannot be 8-arc transitive. Graphs with polynomial growth are one-ended unless they have linear growth. Besides the situation is even more restricted for graphs with more than one end. Thomassen and Woess in [6] showed that an infinite graph with more than one end cannot be 2-arc transitive unless it is a tree. So the situation for undirected graphs with more than one end can be summarized in: An undirected locally infinite connected graph with more than one end is either 0-, 1- or highly arc transitive (hence a tree). Considering directed graphs the situation is much more complicated. Praeger [3] gave infinite families of finite k-arc transitive digraphs for each positive integer k and each degree. And Mansilla and Serra [2] showed that given an arbitrary regular finite digraph and an arbitrary positive integer k, there are infinitely many k-arc transitive finite digraphs which cover the original digraph. Knowing the above mentioned results about transitivity in finite graphs and digraphs the following question arises: Do there exist infinite digraphs which are k-arc transitive for each positive integer k but not highly arc transitive? In the directed case, there are highly arc transitive digraphs with more than one end which do not resemble trees, so the situation is really a different one. Moreover, only examples of 0-, 1- or highly arc transitive digraphs with more than one end were known and that originated the following Conjecture 1: Conjecture 1.1 (Seifter, [4]) A connected locally finite digraph with more than one end is either 0-, 1- or highly arc transitive. In this paper we disprove that conjecture and show the first known examples of sharply 2-arc transitive infinite digraphs with more than one end.
2
Basic definitions
A digraph or directed graph Γ = (V, A)(= (V (Γ), A(Γ))) consists of a set V of vertices and a subset A of ordered pairs from V , called arcs. If (x, y) ∈ A is an arc from x to y, we say that x is adjacent to y and also that y is adjacent − from x. The in-neighbourhood of a vertex x, denoted by NΓ (x), is the set of − vertices adjacent to x, that is, NΓ (x) = {y ∈ V |(y, x) ∈ A}. Furthermore, the in-degree of x, denoted by d+ the cardinality of its in-neighbourhood. Γ (x), is + Similarly, the out-neighbourhood, NΓ (x), and the out-degree, d+ Γ (x), of a vertex x are defined. A digraph is regular of degree r or r-regular when all vertices have in-degree and out-degree r.
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If not stated otherwise, digraphs considered in this paper are infinite, locally finite and connected, that is, all vertices have both finite in- and outdegrees and there is a walk (not necessarily directed) from each given vertex to any other one in the digraph. By Aut Γ we denote the automorphism group of Γ. If Aut Γ acts transitively on V then all vertices have the same in-degree, denoted by d− , and the same Γ out-degree, d+ . For a positive integer k, a k-arc of Γ is a sequence (x0 , ..., xk ) Γ of k +1 vertices such that, for each 0 ≤ i < k, (xi , xi+1 ) is an arc and xi = xi+1 . A digraph Γ is k-arc transitive if it has an automorphism group G < Aut Γ which acts transitively on k-arcs. Further, Γ is sharply k-arc transitive if it is k- but not (k + 1)-arc transitive. Also, Γ is said to be highly arc transitive if it is k-arc transitive for all finite k ≥ 0. And Γ is usually called (vertex) transitive if Aut Γ acts transitively on V . Highly arc transitivity in digraphs is closely related to the so-called property Z. (A digraph Γ has property Z if it exists a digraph homomorphism from Γ onto the directed infinite line which is a Cayley digraph of the additive group of integers.) The concepts of ends can be defined in several different ways; the definition we are going to use is due to Dunwoody [1]. We say that an infinite digraph Γ has more than one end if it has a finite (arc-) cut with infinite sides. Otherwise, Γ has one end. A cut is said to be tight if it partitions V into connected sides A and B. If F and F are cuts with sides A, B and A , B respectively, then F and F cross if all four A ∩ A , A ∩ B , B ∩ A , B ∩ B are non-empty. Dunwoody proved the following: Theorem 2.1 (Dunwoody, [1]) A digraph with more than one end has a infinite tight cut F such that both sides of F are infinite and such that F crosses no φ(F ), where φ ∈ Aut Γ. A finite tight cut F satisfying the above theorem is said to be a D-cut.
3
An infinite family of sharply 2-arc transitive twoended digraphs
We disclaim in this section Conjecture 1 posed by Seifter in [4], that stated that a connected locally infinite digraph with more than one end is either 0-, 1- or highly arc transitive. In the mentioned paper [4] the author gave the following partial solution to
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S.P. Mansilla / Electronic Notes in Discrete Mathematics 29 (2007) 243–247
the conjecture, for regular digraphs of prime degree with a connected D-cut: Theorem 3.1 (Seifter, [4]) Let Γ be a regular transitive digraph with prime degree and more than one end. If Γ has a connected D-cut, then Γ is either 0-,1- or highly arc transitive. In the following, we describe an infinite family of 2-arc transitive two-ended digraphs, not 3-arc transitive, of degree 2 and property Z. Thus, we disprove Seifter’s Conjecture in the general case, even for prime degree. Nonetheless, the partial solution given by Seifter in the above mentioned theorem is in some sense best possible and hence the existence of a connected D-cut in the digraph essential. Theorem 3.2 Let Γn be the digraph defined by the following vertex-set and arc-set: V (Γn ) = Zn× Zn× Z A(Γn ) = {((i, j, k), (j, i, k + 1)), ((i, j, k), (j, i + 1, k + 1))|(i, j, k) ∈ V (Γn )} for n ≥ 3. Then, Γn is a connected 2-regular sharply 2-arc transitive digraph with property Z. See in Figure 1 below a partial depiction of Γ4 .
Fig. 1. Γ4
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Acknowledgement Part of this work was done during a four month stay at the Montanuniversit¨at in Leoben, Austria. The author wants to thank Norbert Seifter for the helpful discussions. This visit was funded by the Generalitat de Catalunya under the mobility grant BE 00512.
References [1] M. J. Dunwoody, Cutting up graphs, Combinatorica 2 (1982) 15–23. [2] S.P. Mansilla, O. Serra, Construction of k-arc transitive digraphs, Discrete Math. 231 (2001) 337–349. [3] C.E. Praeger, Highly arc transitive digraphs, European J. Combin. 10 (1989) 281–292. [4] N. Seifter, Transitive digraphs with more than one end, Discrete Math. to appear. [5] N. Seifter, Properties of graphs with polynomial growth, J. Combin. Theory Ser. B 52 (1991) 222–235. [6] C. Thomassen, W. Woess, Vertex-transitive graphs and accessibility, J. Combin. Theory Ser. B 58 (1993) 248–268. [7] W.T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959) 621– 624. [8] R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica 1 (1981) 309– 311.
An infinite family of sharply two-arc transitive digraphs S`onia P. Mansilla 1,2 Departament de Matem` atica Aplicada IV Universitat Polit`ecnica de Catalunya Av. del Canal Ol´ımpic, s/n, 08860 Castelldefels, Spain
Abstract We disclaim Conjecture 1 posed by Seifter in [N. Seifter, Transitive digraphs with more than one end, Discrete Math., to appear], that stated that a connected locally finite digraph with more than one end is either 0-, 1- or highly arc transitive. We describe in this work an infinite family of 2-arc transitive two-ended digraphs, that are not 3-arc transitive. Keywords: Highly arc transitive, k-arc transitive digraphs, ends of graphs.
1
Introduction
Since decades transitive graphs have been the subject of much investigation. The study of k-edge transitive (undirected) graphs goes back to Tutte [7], who showed that finite cubic graphs cannot be k-edge transitive for k > 5. Weiss [8] proved several years later that the only finite connected k-edge transitive 1
Work supported partly by the Spanish Research Council (Comisi´ on Interministerial de Ciencia y Tecnolog´ıa) under the project MTM2005-08990-C02-01. 2 Email: [email protected] 1571-0653/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.endm.2007.07.041
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S.P. Mansilla / Electronic Notes in Discrete Mathematics 29 (2007) 243–247
graphs with k ≥ 8 are the cycles. In 1991 this result was extended to infinite graphs of polynomial growth: Seifter in [5] proved that a connected, vertextransitive graph of polynomial growth with vertex degrees at least 3 cannot be 8-arc transitive. Graphs with polynomial growth are one-ended unless they have linear growth. Besides the situation is even more restricted for graphs with more than one end. Thomassen and Woess in [6] showed that an infinite graph with more than one end cannot be 2-arc transitive unless it is a tree. So the situation for undirected graphs with more than one end can be summarized in: An undirected locally infinite connected graph with more than one end is either 0-, 1- or highly arc transitive (hence a tree). Considering directed graphs the situation is much more complicated. Praeger [3] gave infinite families of finite k-arc transitive digraphs for each positive integer k and each degree. And Mansilla and Serra [2] showed that given an arbitrary regular finite digraph and an arbitrary positive integer k, there are infinitely many k-arc transitive finite digraphs which cover the original digraph. Knowing the above mentioned results about transitivity in finite graphs and digraphs the following question arises: Do there exist infinite digraphs which are k-arc transitive for each positive integer k but not highly arc transitive? In the directed case, there are highly arc transitive digraphs with more than one end which do not resemble trees, so the situation is really a different one. Moreover, only examples of 0-, 1- or highly arc transitive digraphs with more than one end were known and that originated the following Conjecture 1: Conjecture 1.1 (Seifter, [4]) A connected locally finite digraph with more than one end is either 0-, 1- or highly arc transitive. In this paper we disprove that conjecture and show the first known examples of sharply 2-arc transitive infinite digraphs with more than one end.
2
Basic definitions
A digraph or directed graph Γ = (V, A)(= (V (Γ), A(Γ))) consists of a set V of vertices and a subset A of ordered pairs from V , called arcs. If (x, y) ∈ A is an arc from x to y, we say that x is adjacent to y and also that y is adjacent − from x. The in-neighbourhood of a vertex x, denoted by NΓ (x), is the set of − vertices adjacent to x, that is, NΓ (x) = {y ∈ V |(y, x) ∈ A}. Furthermore, the in-degree of x, denoted by d+ the cardinality of its in-neighbourhood. Γ (x), is + Similarly, the out-neighbourhood, NΓ (x), and the out-degree, d+ Γ (x), of a vertex x are defined. A digraph is regular of degree r or r-regular when all vertices have in-degree and out-degree r.
S.P. Mansilla / Electronic Notes in Discrete Mathematics 29 (2007) 243–247
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If not stated otherwise, digraphs considered in this paper are infinite, locally finite and connected, that is, all vertices have both finite in- and outdegrees and there is a walk (not necessarily directed) from each given vertex to any other one in the digraph. By Aut Γ we denote the automorphism group of Γ. If Aut Γ acts transitively on V then all vertices have the same in-degree, denoted by d− , and the same Γ out-degree, d+ . For a positive integer k, a k-arc of Γ is a sequence (x0 , ..., xk ) Γ of k +1 vertices such that, for each 0 ≤ i < k, (xi , xi+1 ) is an arc and xi = xi+1 . A digraph Γ is k-arc transitive if it has an automorphism group G < Aut Γ which acts transitively on k-arcs. Further, Γ is sharply k-arc transitive if it is k- but not (k + 1)-arc transitive. Also, Γ is said to be highly arc transitive if it is k-arc transitive for all finite k ≥ 0. And Γ is usually called (vertex) transitive if Aut Γ acts transitively on V . Highly arc transitivity in digraphs is closely related to the so-called property Z. (A digraph Γ has property Z if it exists a digraph homomorphism from Γ onto the directed infinite line which is a Cayley digraph of the additive group of integers.) The concepts of ends can be defined in several different ways; the definition we are going to use is due to Dunwoody [1]. We say that an infinite digraph Γ has more than one end if it has a finite (arc-) cut with infinite sides. Otherwise, Γ has one end. A cut is said to be tight if it partitions V into connected sides A and B. If F and F are cuts with sides A, B and A , B respectively, then F and F cross if all four A ∩ A , A ∩ B , B ∩ A , B ∩ B are non-empty. Dunwoody proved the following: Theorem 2.1 (Dunwoody, [1]) A digraph with more than one end has a infinite tight cut F such that both sides of F are infinite and such that F crosses no φ(F ), where φ ∈ Aut Γ. A finite tight cut F satisfying the above theorem is said to be a D-cut.
3
An infinite family of sharply 2-arc transitive twoended digraphs
We disclaim in this section Conjecture 1 posed by Seifter in [4], that stated that a connected locally infinite digraph with more than one end is either 0-, 1- or highly arc transitive. In the mentioned paper [4] the author gave the following partial solution to
246
S.P. Mansilla / Electronic Notes in Discrete Mathematics 29 (2007) 243–247
the conjecture, for regular digraphs of prime degree with a connected D-cut: Theorem 3.1 (Seifter, [4]) Let Γ be a regular transitive digraph with prime degree and more than one end. If Γ has a connected D-cut, then Γ is either 0-,1- or highly arc transitive. In the following, we describe an infinite family of 2-arc transitive two-ended digraphs, not 3-arc transitive, of degree 2 and property Z. Thus, we disprove Seifter’s Conjecture in the general case, even for prime degree. Nonetheless, the partial solution given by Seifter in the above mentioned theorem is in some sense best possible and hence the existence of a connected D-cut in the digraph essential. Theorem 3.2 Let Γn be the digraph defined by the following vertex-set and arc-set: V (Γn ) = Zn× Zn× Z A(Γn ) = {((i, j, k), (j, i, k + 1)), ((i, j, k), (j, i + 1, k + 1))|(i, j, k) ∈ V (Γn )} for n ≥ 3. Then, Γn is a connected 2-regular sharply 2-arc transitive digraph with property Z. See in Figure 1 below a partial depiction of Γ4 .
Fig. 1. Γ4
S.P. Mansilla / Electronic Notes in Discrete Mathematics 29 (2007) 243–247
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Acknowledgement Part of this work was done during a four month stay at the Montanuniversit¨at in Leoben, Austria. The author wants to thank Norbert Seifter for the helpful discussions. This visit was funded by the Generalitat de Catalunya under the mobility grant BE 00512.
References [1] M. J. Dunwoody, Cutting up graphs, Combinatorica 2 (1982) 15–23. [2] S.P. Mansilla, O. Serra, Construction of k-arc transitive digraphs, Discrete Math. 231 (2001) 337–349. [3] C.E. Praeger, Highly arc transitive digraphs, European J. Combin. 10 (1989) 281–292. [4] N. Seifter, Transitive digraphs with more than one end, Discrete Math. to appear. [5] N. Seifter, Properties of graphs with polynomial growth, J. Combin. Theory Ser. B 52 (1991) 222–235. [6] C. Thomassen, W. Woess, Vertex-transitive graphs and accessibility, J. Combin. Theory Ser. B 58 (1993) 248–268. [7] W.T. Tutte, On the symmetry of cubic graphs, Canad. J. Math. 11 (1959) 621– 624. [8] R. Weiss, The nonexistence of 8-transitive graphs, Combinatorica 1 (1981) 309– 311.