On G-locally primitive graphs of locally Twisted Wreath type and a conjecture of Weiss

Let Γ be a connected G-vertex-transitive graph and let v be a vertex of Γ. The graph Γ is said to be G-locally primitive if the action of the vertex-stabiliser G v on the neighbourhood Γ ( v ) of v is primitive. Furthermore, Γ is said to be of locally Twisted Wreath type if G v is a primitive group of Twisted Wreath type in its action on Γ ( v ) . d Weiss conjectured in 1978 that, there exists a function f : N → N such that if Γ is a connected G-vertex-transitive locally primitive graph of valency d and v is a vertex of Γ, then | G v | ⩽ f ( d ) . In this paper we prove this conjecture when Γ is of locally Twisted Wreath type.