Proper conflict-free degree-choosability of outerplanar graphs

A proper coloring ϕ of G is called a proper conflict-free coloring of G if for every non-isolated vertex v of G, there is a color c such that |ϕ^-1 (c) NG (v) |=1. As an analogy to degree-choosability of graphs, the authors recently, in a previous paper, introduced the notion of proper conflict-free (degree+k) -choosability of graphs. For a non-negative integer k, a graph G is proper conflict-free (degree+k) -choosable if for any list assignment L of G with |L (v) | dG (v) +k for every vertex v V (G), G admits a proper conflict-free coloring ϕ such that ϕ (v) L (v) for every vertex v V (G). In this paper, we show that every connected outerplanar graph other than the 5-cycle is proper conflict-free (degree+2) -choosable. This bound is tight in the sense that there are infinitely many connected outerplanar graphs that are not proper conflict-free (degree+1) -choosable. We conclude the paper with two questions for further work.