On 4-Valent Symmetric Graphs

Let G act transitively on incident vertex, edge pairs of the connected 4-valent graph Γ. If a normal subgroup N does not give rise to a natural 4-valent quotient ΓN with G/N acting transitively on incident vertex, edge pairs, then either (a) N has just one or two orbits on vertices, or (b) N has r ⩾ 3 orbits on vertices and the natural quotient ΓN is a circuit Cr (Theorem 1.1). We give a complete classification of the graphs arising in (a) when the normal subgroup N is elementary abelian (Theorems 1.2 and 1.3). Case (b), which depends to some extent on case (a), is more technical and is studied in a subsequent paper.