Free Choosability of Outerplanar Graphs

A graph G is free (a, b)-choosable if for any vertex v with b colors assigned and for any list of colors of size a associated with each vertex u not equal v , the coloring can be completed by choosing for u a subset of b colors such that adjacent vertices are colored with disjoint color sets. In this note, a necessary and sufficient condition for a cycle to be free (a, b)-choosable is given. As a corollary, we obtain almost optimal results about the free (a, b)-choosability of outerplanar graphs.