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 Lv of at least k colors and for any adjacent pair xy, the lists Lx and Ly 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.