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))

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.