Existing poly(d ε^{-1})-query partition oracles for minor-closed bounded-degree graphs can be implemented with poly(d ε^{-1}) log n random bits; any such oracle for cycles requires ω(1) bits when vertices have labels from [N] with N ≥ n.
Lipton and Robert Endre Tarjan
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Reducing the Randomness in Partition Oracles for Bounded Degree Minor-Free Graphs
Existing poly(d ε^{-1})-query partition oracles for minor-closed bounded-degree graphs can be implemented with poly(d ε^{-1}) log n random bits; any such oracle for cycles requires ω(1) bits when vertices have labels from [N] with N ≥ n.