20.500.12556/RUL-179008 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 Angleški jezik Angleški jezik Članek v reviji 2026-02-03 09:10:03 2026-02-03 09:10:04 2026-02-04 04:36:58 0000-00-00 00:00:00 2026 0 0 5 str. iss. 6, art. no. 115003 Vol. 349 Jun. 2026 0000-00-00 Zaloznikova NiDoloceno NiDoloceno 0000-00-00 0000-00-00 2026-06-01 519.17 0012-365X 10.1016/j.disc.2026.115003 266959619 1-s2.0-S0012365X26000270-main.pdf 1-s2.0-S0012365X26000270-main.pdf 1 1A1739287DF4F766C9F1AEA82FF01A54 1fa3e6feb150f3f0ae35dfa8a9ff792d99d06681167804bf65767b5427677485 972f52e1-00d8-11f1-a1ba-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=227068 https://www.sciencedirect.com/science/article/pii/S0012365X26000270 1 acdcde66-00d7-11f1-a1ba-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=227063 Fakulteta za matematiko in fiziko 0 0 0