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 x y , 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.Our strengthening is motivated by the observation that in (4, 1)-list assignment, vertices of an edge have together at least 7 colors, while in (3, 1)-list assignment, they have only at least 5. Our setting gives at least 6 colors.