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We study finite four-valent graphs $$\\Gamma $$Γ admitting an edge-transitive group G of automorphisms such that G determines and preserves an edge-orientation on $$\\Gamma $$Γ, and such that at least one G-normal quotient is a cycle (a quotient modulo the orbits of a normal subgroup of G). We show, on the one hand, that the number of distinct cyclic G-normal quotients can be unboundedly large. On the other hand, existence of independent cyclic G-normal quotients (that is, they are not extendable to a common cyclic G-normal quotient) places severe restrictions on the graph $$\\Gamma $$Γ and we classify all examples. We show there are five infinite families of such pairs $$(\\Gamma ,G)$$(Γ,G) and in particular that all such graphs have at least one normal quotient which is an unoriented cycle. We compare this new approach with existing treatments for the sub-class of weak metacirculant graphs with these properties, finding that only two infinite families of examples occur in common from both analyses. Several open problems are posed.

Finite edge-transitive oriented graphs of valency four with cyclic normal quotients