Abstract
A graph is called a weakmetacirculant if it contains a metacyclic vertextransitive automorphism group, introduced by Marušič and Šparl [5]. It is a generalization of the concept of matecirculant (introduced in [1]), which has a metacyclic vertextransitive automorphism group with certain restrictive condition. A natural question is whether a weakmetacirculant 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 nonsplit metacyclic pgroups, with p odd prime. However, there is, unfortunately, a mistake in [4, Theorem 1.3] which claimed that such Cayley graphs could be edgetransitive. The following theorem corrects the mistake and confirms the statement that such graphs are weakmetacirculants but not metacirculants is still valid. Theorem 1 Let p be an odd prime, and let G be a nonsplit metacyclic pgroup. Let Γ be a connected Cayley graph of G of valency at most 2(p−1). Then AutΓ=G, and Γ is not edgetransitive; in particular, Γ is a weakmetacirculant but not a metacirculant. Proof Let Γ=Cay(G,S). Then Γ is a weakmetacirculant. 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 nonsplit metacyclic pgroup, Aut(G) is a pgroup 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 edgetransitive. Since G is nonsplit, Γ is not a metacirculant, see [4] or [5]. □ It is remarkable that, for a nonsplit metacyclic pgroup G with p being odd and prime, all connected Cayley graphs of G of valency at most 2(p−1) are always socalled graphical regular representations of the group G, and are not edgetransitive.
Original language  English 

Pages (fromto)  344345 
Number of pages  2 
Journal  Journal of Combinatorial Theory. Series A 
Volume  146 
DOIs 

Publication status  Published  1 Feb 2017 
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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)). / Li, Cai Heng; Song, Shu Jiao.
In: Journal of Combinatorial Theory. Series A, Vol. 146, 01.02.2017, p. 344345.Research output: Contribution to journal › Comment/debate
TY  JOUR
T1  Erratum
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 weakmetacirculant if it contains a metacyclic vertextransitive automorphism group, introduced by Marušič and Šparl [5]. It is a generalization of the concept of matecirculant (introduced in [1]), which has a metacyclic vertextransitive automorphism group with certain restrictive condition. A natural question is whether a weakmetacirculant 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 nonsplit metacyclic pgroups, with p odd prime. However, there is, unfortunately, a mistake in [4, Theorem 1.3] which claimed that such Cayley graphs could be edgetransitive. The following theorem corrects the mistake and confirms the statement that such graphs are weakmetacirculants but not metacirculants is still valid. Theorem 1 Let p be an odd prime, and let G be a nonsplit metacyclic pgroup. Let Γ be a connected Cayley graph of G of valency at most 2(p−1). Then AutΓ=G, and Γ is not edgetransitive; in particular, Γ is a weakmetacirculant but not a metacirculant. Proof Let Γ=Cay(G,S). Then Γ is a weakmetacirculant. 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 nonsplit metacyclic pgroup, Aut(G) is a pgroup 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 edgetransitive. Since G is nonsplit, Γ is not a metacirculant, see [4] or [5]. □ It is remarkable that, for a nonsplit metacyclic pgroup G with p being odd and prime, all connected Cayley graphs of G of valency at most 2(p−1) are always socalled graphical regular representations of the group G, and are not edgetransitive.
AB  A graph is called a weakmetacirculant if it contains a metacyclic vertextransitive automorphism group, introduced by Marušič and Šparl [5]. It is a generalization of the concept of matecirculant (introduced in [1]), which has a metacyclic vertextransitive automorphism group with certain restrictive condition. A natural question is whether a weakmetacirculant 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 nonsplit metacyclic pgroups, with p odd prime. However, there is, unfortunately, a mistake in [4, Theorem 1.3] which claimed that such Cayley graphs could be edgetransitive. The following theorem corrects the mistake and confirms the statement that such graphs are weakmetacirculants but not metacirculants is still valid. Theorem 1 Let p be an odd prime, and let G be a nonsplit metacyclic pgroup. Let Γ be a connected Cayley graph of G of valency at most 2(p−1). Then AutΓ=G, and Γ is not edgetransitive; in particular, Γ is a weakmetacirculant but not a metacirculant. Proof Let Γ=Cay(G,S). Then Γ is a weakmetacirculant. 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 nonsplit metacyclic pgroup, Aut(G) is a pgroup 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 edgetransitive. Since G is nonsplit, Γ is not a metacirculant, see [4] or [5]. □ It is remarkable that, for a nonsplit metacyclic pgroup G with p being odd and prime, all connected Cayley graphs of G of valency at most 2(p−1) are always socalled graphical regular representations of the group G, and are not edgetransitive.
UR  http://www.scopus.com/inward/record.url?scp=84997327222&partnerID=8YFLogxK
U2  10.1016/j.jcta.2016.09.005
DO  10.1016/j.jcta.2016.09.005
M3  Comment/debate
VL  146
SP  344
EP  345
JO  Journal of Combinatorial Theory Series A
JF  Journal of Combinatorial Theory Series A
SN  00219800
ER 