On edge-transitive cubic graphs of square-free order

Abstract

a b s t r a c tA regular graph is said to be semisymmetric if it is edge-transitive but not vertex-transitive. In this paper, we give a complete list of connected semisymmetric cubic graph of square-free order, which consists of one single graph of order 210 and four infinite families of such graphs.

The vertex-transitive and edge-transitive tetravalent graphs of square-free order

“…Applying Lemmas 6.1-6.4 to the pair (X, Γ C ), we conclude that T = soc(X ) = PSL(2, p) and Γ C is X -arc-transitive, and so Γ is arc-transitive. Moreover, G α ∼ = X B ∼ = Z 2 2 , D 8 or D 16 , where B is a C-orbit and α ∈ B. If X B = X α , then G α ≤ X ; in this case, it is easily shown that Γ is isomorphic to the standard double cover of Γ C , and so Theorem 1.1 (10) occurs.…”

“…Since Γ is connected, C fixes each vertex of Γ , and so C = 1 as C ≤ AutΓ , a contradiction. Thus |G αβ | is a divisor of 4, and hence G α ∼ = D 8 or D 16 .…”

“…If X B ∼ = Z 2 2 , then X = PSL(2, p) and Γ is arc-regular, and hence Theorem 1.1 (7) occurs. We next deal with the cases: X B ∼ = D 8 and X B ∼ = D 16 .…”

“…Set G = C×X . Suppose that X has a Sylow 2-subgroup P ∼ = Z 2 2 , D 8 or D 16 . Write P = Q: z , where z is an involution.…”

“…In [16], we gave a characterization for the class of edge-transitive graphs of squarefree order, which says that the basic members in this class consist of a few special families of graphs and a finite number of sporadic graphs. This motivates us to classify edge-transitive graphs of square-free order and of small valency.…”