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Corrigendum to “A characterization of metacirculants” [J. Combin. Theory Ser. A 120 (1) (2013) 39–48]
Journal of Combinatorial Theory, Series A 146 (2017) 344–345
Contents lists available at ScienceDirect
Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta
Corrigendum
Corrigendum to “A characterization of metacirculants” [J. Combin. Theory Ser. A 120 (1) (2013) 39–48] Cai Heng Li, Shu Jiao Song ∗ School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
a r t i c l e
i n f o
Article history: Received 1 September 2015 Available online 24 October 2016
A graph is called a weak-metacirculant if it contains a metacyclic vertex-transitive automorphism group, introduced by Marušič and Šparl [5]. It is a generalization of the concept of matecirculant (introduced in [1]), which has a metacyclic vertex-transitive automorphism group with certain restrictive condition. A natural question is whether a weak-metacirculant is actually a metacirculant, see [5]. The answer to this question is negative: an infinitely family of examples is given in [4, Theorem 1.3], which are tetravalent Cayley graphs of non-split metacyclic p-groups, with p odd prime. However, there is, unfortunately, a mistake in [4, Theorem 1.3] which claimed that such Cayley graphs could be edge-transitive. The following theorem corrects the mistake and confirms the statement that such graphs are weak-metacirculants but not metacirculants is still valid.
DOI of original article: http://dx.doi.org/10.1016/j.jcta.2012.06.010.
* Corresponding author. E-mail addresses: [email protected] (C.H. Li), [email protected] (S.J. Song). http://dx.doi.org/10.1016/j.jcta.2016.09.005 0097-3165/© 2016 Elsevier Inc. All rights reserved.
C.H. Li, S.J. Song / Journal of Combinatorial Theory, Series A 146 (2017) 344–345
345
Theorem 1. Let p be an odd prime, and let G be a non-split metacyclic p-group. Let Γ be a connected Cayley graph of G of valency at most 2(p − 1). Then AutΓ = G, and Γ is not edge-transitive; in particular, Γ is a weak-metacirculant but not a metacirculant. Proof. Let Γ = Cay(G, S). Then Γ is a weak-metacirculant. Let A = AutΓ, let α be the vertex of Γ corresponding to the identity of G. By [3, Corollary 1.2], A = G : Aut(G, S), where Aut(G, S) = σ ∈ Aut(G) | S σ = S ≤ Aut(G). Since G is a non-split metacyclic p-group, Aut(G) is a p-group by [2]. −1 −1 Let S = {s1 , s−1 1 , s2 , . . . , sk , sk }, where k ≤ p − 1. Suppose that Aut(G, S) = 1. Then there exists an element σ ∈ Aut(G, S) of order p. As |S| = 2k < 2p, it implies i that sσj = s−1 for some integers i and j, which is not possible as p is odd. Thus j Aα = Aut(G, S) = 1, and A = G. In particular, Γ is not edge-transitive. Since G is non-split, Γ is not a metacirculant, see [4] or [5]. 2 It is remarkable that, for a non-split metacyclic p-group G with p being odd and prime, all connected Cayley graphs of G of valency at most 2(p − 1) are always so-called graphical regular representations of the group G, and are not edge-transitive. References [1] B. Alspach, T.D. Parsons, A construction for vertex-transitive graphs, Canad. J. Math. 34 (1982) 307–318. [2] M.J. Curran, The automorphism group of a nonsplit metacyclic p-group, Arch. Math. 90 (2008) 483–489. [3] C.H. Li, H.S. Sim, Automorphisms of metacyclic Cayley graphs of prime-power order, J. Aust. Math. Soc. 71 (2001) 223–233. [4] C.H. Li, S.J. Song, D.J. Wang, A characterization of metacirculants, J. Combin. Theory Ser. A 120 (2013) 39–48. [5] D. Marušič, P. Šparl, On quartic half-arc-transitive metacirculants, J. Algebraic Combin. 28 (2008) 365–395.
Contents lists available at ScienceDirect
Journal of Combinatorial Theory, Series A www.elsevier.com/locate/jcta
Corrigendum
Corrigendum to “A characterization of metacirculants” [J. Combin. Theory Ser. A 120 (1) (2013) 39–48] Cai Heng Li, Shu Jiao Song ∗ School of Mathematics and Statistics, The University of Western Australia, Crawley, WA 6009, Australia
a r t i c l e
i n f o
Article history: Received 1 September 2015 Available online 24 October 2016
A graph is called a weak-metacirculant if it contains a metacyclic vertex-transitive automorphism group, introduced by Marušič and Šparl [5]. It is a generalization of the concept of matecirculant (introduced in [1]), which has a metacyclic vertex-transitive automorphism group with certain restrictive condition. A natural question is whether a weak-metacirculant is actually a metacirculant, see [5]. The answer to this question is negative: an infinitely family of examples is given in [4, Theorem 1.3], which are tetravalent Cayley graphs of non-split metacyclic p-groups, with p odd prime. However, there is, unfortunately, a mistake in [4, Theorem 1.3] which claimed that such Cayley graphs could be edge-transitive. The following theorem corrects the mistake and confirms the statement that such graphs are weak-metacirculants but not metacirculants is still valid.
DOI of original article: http://dx.doi.org/10.1016/j.jcta.2012.06.010.
* Corresponding author. E-mail addresses: [email protected] (C.H. Li), [email protected] (S.J. Song). http://dx.doi.org/10.1016/j.jcta.2016.09.005 0097-3165/© 2016 Elsevier Inc. All rights reserved.
C.H. Li, S.J. Song / Journal of Combinatorial Theory, Series A 146 (2017) 344–345
345
Theorem 1. Let p be an odd prime, and let G be a non-split metacyclic p-group. Let Γ be a connected Cayley graph of G of valency at most 2(p − 1). Then AutΓ = G, and Γ is not edge-transitive; in particular, Γ is a weak-metacirculant but not a metacirculant. Proof. Let Γ = Cay(G, S). Then Γ is a weak-metacirculant. Let A = AutΓ, let α be the vertex of Γ corresponding to the identity of G. By [3, Corollary 1.2], A = G : Aut(G, S), where Aut(G, S) = σ ∈ Aut(G) | S σ = S ≤ Aut(G). Since G is a non-split metacyclic p-group, Aut(G) is a p-group by [2]. −1 −1 Let S = {s1 , s−1 1 , s2 , . . . , sk , sk }, where k ≤ p − 1. Suppose that Aut(G, S) = 1. Then there exists an element σ ∈ Aut(G, S) of order p. As |S| = 2k < 2p, it implies i that sσj = s−1 for some integers i and j, which is not possible as p is odd. Thus j Aα = Aut(G, S) = 1, and A = G. In particular, Γ is not edge-transitive. Since G is non-split, Γ is not a metacirculant, see [4] or [5]. 2 It is remarkable that, for a non-split metacyclic p-group G with p being odd and prime, all connected Cayley graphs of G of valency at most 2(p − 1) are always so-called graphical regular representations of the group G, and are not edge-transitive. References [1] B. Alspach, T.D. Parsons, A construction for vertex-transitive graphs, Canad. J. Math. 34 (1982) 307–318. [2] M.J. Curran, The automorphism group of a nonsplit metacyclic p-group, Arch. Math. 90 (2008) 483–489. [3] C.H. Li, H.S. Sim, Automorphisms of metacyclic Cayley graphs of prime-power order, J. Aust. Math. Soc. 71 (2001) 223–233. [4] C.H. Li, S.J. Song, D.J. Wang, A characterization of metacirculants, J. Combin. Theory Ser. A 120 (2013) 39–48. [5] D. Marušič, P. Šparl, On quartic half-arc-transitive metacirculants, J. Algebraic Combin. 28 (2008) 365–395.