Tetravalent Cayley Graphs Of Groups Of Order 3p～2

Let G be a finite group and S a subset of G such that 1 (?) S. The Cayley (di) graph X = Cay(G, S) on G relative to S is defined to have vertex set V(X) = G and edge set E(X) = {(g,sg)|g∈G,s∈S}. A Cayley graph X = Cay(G,S) is said to be normal if R(G) is normal in AutX. Let p be a prime and p > 3, G = p = c^p = b³ = 1 = [a, c], a^b = c, c^b = a^-1c^-1>, and H = p² = b³ = 1,a^b = a^r>, where r (?) 1(mod p²), r³â‰¡1(mod p²), 3|(p—1). In this paper ,we prove that all tetravalent Cayley graphs of G and H are normal by a combination of group theoretical and combinatorial method. As a byproduct, we obtain an infinite family of 4-valent one-regular graphs of order 3p².