Isomorphisms of the 4-valent generalized Cayley graphs of Z3p

For an odd prime p ≠ 3, the cyclic group Z3p≅Z3 × Zp, where Z3=〈w1〉 and Zp=〈w2〉. Let β1 and β2 be involutory automorphisms of Z3p defined by (w1,w2)β1=(w1−1,w2) and (w1,w2)β2=(w1,w2−1). Then the elements of sets Δ1={{(s1,s2),(s1,s2−1),(s1,s3),(s1,s3−1)}∣s1∈Z3,s2≠s3ands2,s3≠e2} and Δ2={{(w1,t1),(w1−1,t1),(w1,t2),(w1−1,t2)}∣t1,t2∈Zpandt1≠t2} form generalized Cayley subsets of Z3p with respect to β1 and β2, respectively. In this paper, we prove that for i=1 and 2, if S, T ∈ Δi, then GC(Z3p, S, βi) and GC(Z3p, T, βi) are isomorphic if and only if they are generalized Cayley isomorphic.