Asymmetrizing cost and density of vertex-transitive cubic graphs

A set S of vertices in a graph G with nontrivial automorphism group is asymmetrizing if the identity mapping is the only automorphism of G that preserves S as a set. If such sets exist, then their minimum cardinality is the asymmetrizing cost ρ ( G ) of G . For finite graphs the asymmetrizing density δ ( G ) of G is the quotient of the size of S by the order of G . For infinite graphs δ ( G ) is defined by a limit process. Many classes of infinite graphs with δ ( G ) = 0 are known, but seemingly no infinite vertex transitive graphs with δ ( G ) > 0. Here, we construct connected, infinite vertex transitive cubic graphs of asymmetrizing density δ ( G ) = 1 n 2 n +1 for each n ≥ 1. We also construct finite vertex transitive cubic graphs of arbitrarily large asymmetrizing cost. The examples are Split Praeger–Xu graphs, for which we provide another characterization. This contrasts with our results for vertex transitive cubic graphs that have one arc orbit or are so-called synchronously connected graphs with two arc orbits. For them we show that ρ ( G ) is either ≤ 5 or infinite. In the latter case δ ( G ) = 0.