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 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 ∈ N. The following results make partial progress towards this conjecture. Strengthening results of Tuza and Voigt, and of Schauz, we prove for any m ∈ 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 ∈ N, G is (∆(G)m, m)-paintable.
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