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A construction is given for an infinite family{Γ_n} of finite vertex-transitive non-Cayley graphs offixed valency with the property that the order of the vertex-stabilizer inthe smallest vertex-transitive group of automorphisms ofΓ_n is a strictly increasing function ofn. For each n the graph is 4-valent and arc-transitive, with automorphism group a symmetric group of largeprime degree p2^{2^n+2}. The construction uses Sierpinski‘sgasket to produce generating permutations for the vertex-stabilizer (alarge 2-group).

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