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 Gv on the neighbourhood γ(v) of v is primitive. Furthermore, γ is said to be of locally Twisted Wreath type if Gv is a primitive group of Twisted Wreath type in its action on γ(v).Richard 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 |Gv|≤f(d). In this paper we prove this conjecture when γ is of locally Twisted Wreath type. © 2011 Elsevier Inc.