Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy

Kashima, Masaki (author), Škrekovski, Riste (author), Xu, Rongxing (author)

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.

• Language: English
• Keywords: proper conflict-free coloring, list coloring, degree-choosability, degeneracy
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FMF - Faculty of Mathematics and Physics
• Publication version: Version of Record
• Publication date: 01.06.2026
• Year: 2026
• Number of pages: 5 str.
• Numbering: Vol. 349, iss. 6, art. no. 115003
• PID: 20.500.12556/RUL-179008
• UDC: 519.17
• ISSN on article: 0012-365X
• DOI: 10.1016/j.disc.2026.115003
• COBISS.SI-ID: 266959619
• Publication date in RUL: 03.02.2026

Record is a part of a journal

• Title: Discrete mathematics
• Shortened title: Discrete math.
• Publisher: Elsevier
• ISSN: 0012-365X
• COBISS.SI-ID: 1118479

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.