Results on proper conflict-free list coloring of graphs

Given a graph G and a mapping f:V(G) → N, an f-list assignment of G is a function that maps each v ∈ V(G) to a set of at least $f(v)$ colors. For an f-list assignment L of a graph G, a proper conflict-free L-coloring of G is a proper coloring $ϕ$ of G such that ϕ(v) ∈ L(v) for every vertex v∈ V(G) and v has a color that appears precisely once at its neighborhood for every non-isolated vertex v∈ V(G). We say that G is proper conflict-free f-choosable if for any f-list assignment L of G, there exists a proper conflict-free L-coloring of G. For a non-negative integer k, we say that G is proper conflict-free ( degree+k)-choosable if G is proper conflict-free f-choosable where f is a mapping with f(v)= dG(v)+k for every vertex v∈ V(G). Motivated by degree-choosability of graphs, we investigate the proper conflict-free ( degree+k)-choosability of graphs, especially for cases $k=1,2,3$. As the 5-cycle is not proper conflict-free ( degree+2)-choosable and it is the only such graph we know, it is possible that every connected graph other than the 5-cycle is proper conflict-free ( degree+2)-choosable and thus every graph is proper conflict-free ( degree+3)-choosable. To support these, we show that every connected graph with maximum degree at most 3 distinct from the 5-cycle is proper conflict-free (degree+2)-choosable, and that $S(G)$ is proper conflict-free (degree+2)-choosable for every graph G, where $S(G)$ is a graph obtained from G by subdividing each edge once. Furthermore, by adapting the technique of DP-colorings, we prove that every graph with maximum degree at most $4$ is proper conflict-free ( degree+3)-choosable.