Erratum: Corrigendum to “A characterization of metacirculants” (Journal of Combinatorial Theory. Series A (2013) 1 (39–48)(S0097316512001148)(10.1016/j.jcta.2012.06.010))

Cai Heng Li, Shu Jiao Song

    Research output: Contribution to journalComment/debate

    2 Citations (Scopus)

    Abstract

    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. 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)|Sσ=S〉≤Aut(G). Since G is a non-split metacyclic p-group, Aut(G) is a p-group by [2]. Let S={s1,s1−1,s2−1,…,sk,sk−1}, 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 that sjσi=sj−1 for some integers i and j, which is not possible as p is odd. Thus 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].  □ 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.

    Original languageEnglish
    Pages (from-to)344-345
    Number of pages2
    JournalJournal of Combinatorial Theory. Series A
    Volume146
    DOIs
    Publication statusPublished - 1 Feb 2017

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    @article{d238fb60d4ba446a8efa3df474532b08,
    title = "Erratum: Corrigendum to “A characterization of metacirculants” (Journal of Combinatorial Theory. Series A (2013) 1 (39–48)(S0097316512001148)(10.1016/j.jcta.2012.06.010))",
    abstract = "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. 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)|Sσ=S〉≤Aut(G). Since G is a non-split metacyclic p-group, Aut(G) is a p-group by [2]. Let S={s1,s1−1,s2−1,…,sk,sk−1}, 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 that sjσi=sj−1 for some integers i and j, which is not possible as p is odd. Thus 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].  □ 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.",
    author = "Li, {Cai Heng} and Song, {Shu Jiao}",
    year = "2017",
    month = "2",
    day = "1",
    doi = "10.1016/j.jcta.2016.09.005",
    language = "English",
    volume = "146",
    pages = "344--345",
    journal = "Journal of Combinatorial Theory Series A",
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    T2 - Corrigendum to “A characterization of metacirculants” (Journal of Combinatorial Theory. Series A (2013) 1 (39–48)(S0097316512001148)(10.1016/j.jcta.2012.06.010))

    AU - Li, Cai Heng

    AU - Song, Shu Jiao

    PY - 2017/2/1

    Y1 - 2017/2/1

    N2 - 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. 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)|Sσ=S〉≤Aut(G). Since G is a non-split metacyclic p-group, Aut(G) is a p-group by [2]. Let S={s1,s1−1,s2−1,…,sk,sk−1}, 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 that sjσi=sj−1 for some integers i and j, which is not possible as p is odd. Thus 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].  □ 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.

    AB - 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. 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)|Sσ=S〉≤Aut(G). Since G is a non-split metacyclic p-group, Aut(G) is a p-group by [2]. Let S={s1,s1−1,s2−1,…,sk,sk−1}, 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 that sjσi=sj−1 for some integers i and j, which is not possible as p is odd. Thus 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].  □ 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.

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    JO - Journal of Combinatorial Theory Series A

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