Vertex-Transitive Non-Cayley Graphs with Arbitrarily Large Vertex-Stabilizer

Abstract

A construction is given for an infinite family {Γn} of finite vertex-transitive non-Cayley graphs of fixed valency with the property that the order of the vertex-stabilizer in the smallest vertex-transitive group of automorphisms of Γn is a strictly increasing function ofn . For each n the graph is 4-valent and arc-transitive, with automorphism group a symmetric group of large prime degree \(p < 2^{2^n + 2}\). The construction uses Sierpinski's gasket to produce generating permutations for the vertex-stabilizer (a large 2-group).