{"abstract":"Let a, b \u2208 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) \u2286 L(v) and C(u) \u2229 C(w) = \u2205 for any uw \u2208 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 each vertex v is allowed to be marked a times, there is an algorithm to produce a coloring in which each vertex v receives b colors such that adjacent vertices receive disjoint sets of colors. We show that every odd cycle C 2k+1 is (a, b)-paintable exactly when it is (a, b)-chosable, which is when a 2b + b/k. In 2009, Zhu conjectured that if G is (a, 1)-paintable, then G is (am, m)-paintable for any m \u2208 N. The following results make partial progress towards this conjecture. Strengthening results of Tuza and Voigt, and of Schauz, we prove for any m \u2208 N that G is (5m, m)-paintable when G is planar. Strengthening work of Tuza and Voigt, and of Hladky, Kral, and Schauz, we prove that for any connected graph G other than an odd cycle or complete graph and any m \u2208 N, G is (\u2206(G)m, m)-paintable.","author":[{"family":"Mahoney"}],"id":"unknown","issued":{"date-parts":[[2017]]},"title":"Strengthening (a, b)-Choosability Results to (a, b)-Paintability","type":"article-journal"}