On Choosability with Separation of Planar Graphs with Forbidden Cycles
Ilkyoo Choi, Bernard Lidický, Derrick Stolee · Journal of Graph Theory · 2015
Abstract We study choosability with separation which is a constrained version of list coloring of graphs. A ‐list assignment L of a graph G is a function that assigns to each vertex v a list of at least k colors and for any adjacent pair , the lists and share at most d colors. A graph G is ‐choosable if there exists an L‐coloring of G for every ‐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.