20.500.12556/DiRROS-27407 Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy A 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. proper conflict-free coloring list coloring degree-choosability degeneracy true false true Elsevier Angleški jezik Ni določen © 2026 The Authors Neznano 2026-02-05 15:39:34 2026-02-05 15:39:35 2026-02-06 03:51:21 0000-00-00 00:00:00 2026 0 0 5 str. iss. 6, [article no.] 115003 Vol. 349 Jun. 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-06-01 519.17 0012-365X 10.1016/j.disc.2026.115003 266959619 https://www.sciencedirect.com/science/article/pii/S0012365X26000270 1 7f59b735-02a0-11f1-93b0-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=40092 RAZ_Kashima_Masaki_2026.pdf RAZ_Kashima_Masaki_2026.pdf 1 AD2D3B57BBB88D947CCDB7E69257710D 53aa6d6a662fe97ffb820fe1b921b4098b2229dd4e771b8557d7ca86815a7740 b97babb5-02a0-11f1-93b0-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=40093 Rudolfovo – Znanstveno in tehnološko središče Novo mesto 0 0 0