Imprimitive symmetric graphs with cyclic blocks

Let ΓΓ be a graph admitting an arc-transitive subgroup GG of automorphisms that leaves invariant a vertex partition BB with parts of size v≥3v≥3. In this paper we study such graphs where: for B,C∈BB,C∈B connected by some edge of ΓΓ, exactly two vertices of BB lie on no edge with a vertex of CC; and as CC runs over all parts of BB connected to BB these vertex pairs (ignoring multiplicities) form a cycle. We prove that this occurs if and only if v=3v=3 or 4, and moreover we give three geometric or group theoretic constructions of infinite families of such graphs.