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On Choosability with Separation of Planar Graphs with Forbidden Cycles

We study choosability with separation which is a constrained version of list coloring of graphs. A (k,d) ‐list assignment L of a graph G is a function 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. We prove that planar graphs without 4‐cycles are (3, 1)‐choosable and that planar graphs without 5‐ and 6‐cycles are (3, 1)‐choosable. In addition, we give an alternative and slightly stronger proof that triangle‐free planar graphs are (3, 1)‐choosable.