A class of symmetric graphs with 2‐arc transitive quotients

Let Γ be an X ‐symmetric graph admitting an X ‐invariant partition ℬ on V (Γ) such that Γ ℬ is connected and ( X , 2)‐arc transitive. A characterization of (Γ, X , ℬ) was given in [S. Zhou Eur J Comb 23 (2002), 741–760] for the case where | B |>|Γ( C )∩ B |=2 for an arc ( B, C ) of Γ ℬ .We con‐sider in this article the case where | B |>|Γ( C )∩ B |=3, and prove that Γ can be constructed from a 2‐arc transitive graph of valency 4 or 7 unless its connected components are isomorphic to 3 K 2 , C 6 or K 3, 3 . As a byproduct, we prove that each connected tetravalent ( X , 2)‐transitive graph is either the complete graph K 5 or a near n ‐gonal graph for some n ⩾4. © 2010 Wiley Periodicals, Inc. J Graph Theory 65: 232–245, 2010