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.