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.
Kashima et al. (Thu,) studied this question.