Abstract
Let and . We say is -regular Cayley graph if acts regularly on its arcs. is said to be core-free if is core-free in some . In this paper, we prove that if an -regular Cayley graph of valency is not normal or binormal, then it is the normal cover of one of two core-free ones up to isomorphism. In particular, there are no core-free -regular Cayley graphs of valency .
Highlights
We assume that all graphs in this paper are finite, simple, and undirected.Let Γ be a graph
A graph Γ can be viewed as a Cayley graph of a group G if and only if AutΓ contains a subgroup that is isomorphic to G and acts regularly on the vertex set
The main result of this paper is the following assertion
Summary
We assume that all graphs in this paper are finite, simple, and undirected. Let Γ be a graph. Li proved in [1] that there are only finite number of core-free s-transitive Cayley graphs of valency k for s ∈ {2, 3, 4, 5, 7} and k ≥ 3 and that, with the exceptions s = 2 and (s, k) = (3, 7), every s-transitive Cayley graph is a normal cover of a core-free one. Motivated by above results and problem, we consider 1regular Cayley graphs of valency 5 in this paper. A graph Γ can be viewed as a Cayley graph of a group G if and only if AutΓ contains a subgroup that is isomorphic to G and acts regularly on the vertex set. Let Γ = Cay(G, S) be an (X, 1)-regular Cayley graph of valency 5, where G ≤ X ≤ AutΓ. Let Γ = Cay(G, S) be an 1-regular Cayley graph of valency 5.
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