Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracyKashima, Masaki (Avtor) Škrekovski, Riste (Avtor) Xu, Rongxing (Avtor) proper conflict-free coloringlist coloringdegree-choosabilitydegeneracyA proper coloring ▫$\phi$▫ 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 ▫$|\phi^{-1}(c) \cap N_G(v)| = 1$▫. As an analogy of degree-choosability of graphs, we 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 ▫$|L(v)| \ge d_G(v) + k$▫ for every vertex ▫$v \in V(G)$▫, ▫$G$▫ admits a proper conflict-free coloring ▫$\phi$▫ such that ▫$\phi(v) \in L(v)$▫ for every vertex ▫$v \in V(G)$▫. In this note, we first remark if a graph ▫$G$▫ is ▫$d$▫-degenerate, then ▫$G$▫ is proper conflict-free (degree ▫$+d+1$▫)-choosable. Furthermore, when ▫$d=1$▫, we can reduce the number of colors by showing that every tree is proper conflict-free (degree ▫$+1$▫)-choosable. This motivates us to state a question.Elsevier20262026-02-05 15:39:34Neznano27407sl© 2026 The Authors