On choosability with separation of planar graphs with lists of different sizes

A (k,d)-list assignment L of a graph G is a mapping that assigns to each vertex v a list L(v) of at least k colors and for any adjacent pair xy, the lists L(x) and L(y) share at most d colors. A graph G is (k,d)-choosable if there exists an L-coloring of G for every (k,d)-list assignment L. This concept is also known as choosability with separation. It is known that planar graphs are (4,1)-choosable but it is not known if planar graphs are (3,1)-choosable. We strengthen the result that planar graphs are (4,1)-choosable by allowing an independent set of vertices to have lists of size 3 instead of 4.

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