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Strengthening (a, b)-Choosability Results to (a, b)-Paintability
Let a, b ∈ N. A graph G is (a, b)-choosable if for any list assignment L such that |L(v)| a, there exists a coloring in which each vertex v receives a set C(v) of b colors such that C(v) ⊆ L(v) and C(u) ∩ C(w) = ∅ for any uw ∈ E(G). In the online version of this problem, on each round, a set of vertices allowed to receive a particular color is marked, and the coloring algorithm chooses an independent subset of these vertices to receive that color. We say G is (a, b)-paintable if when eachfatcat:lcboe65g4jbfrlkuwfrabfydhq