Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914046801870848 |
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| author | Kashima, Masaki Škrekovski, Riste Xu, Rongxing |
| author_facet | Kashima, Masaki Škrekovski, Riste Xu, Rongxing |
| contents | A proper coloring $ϕ$ 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 $|ϕ^{-1}(c)\cap N_G(v)|=1$. As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free $({\rm degree}+k)$-choosability of graphs. For a non-negative integer $k$, a graph $G$ is proper conflict-free $({\rm degree}+k)$-choosable if for any list assignment $L$ of $G$ with $|L(v)|\geq d_G(v)+k$ for every vertex $v\in V(G)$, $G$ admits a proper conflict-free coloring $ϕ$ such that $ϕ(v)\in L(v)$ 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 $({\rm degree}+d+1)$-choosable. Furthermore, when $d=1$, we can reduce the number of colors by showing that every tree is proper conflict-free $({\rm degree}+1)$-choosable. This motivates us to state a question. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_12560 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy Kashima, Masaki Škrekovski, Riste Xu, Rongxing Combinatorics 05C15 A proper coloring $ϕ$ 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 $|ϕ^{-1}(c)\cap N_G(v)|=1$. As an analogy of degree-choosability of graphs, we introduced the notion of proper conflict-free $({\rm degree}+k)$-choosability of graphs. For a non-negative integer $k$, a graph $G$ is proper conflict-free $({\rm degree}+k)$-choosable if for any list assignment $L$ of $G$ with $|L(v)|\geq d_G(v)+k$ for every vertex $v\in V(G)$, $G$ admits a proper conflict-free coloring $ϕ$ such that $ϕ(v)\in L(v)$ 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 $({\rm degree}+d+1)$-choosable. Furthermore, when $d=1$, we can reduce the number of colors by showing that every tree is proper conflict-free $({\rm degree}+1)$-choosable. This motivates us to state a question. |
| title | Remarks on proper conflict-free degree-choosability of graphs with prescribed degeneracy |
| topic | Combinatorics 05C15 |
| url | https://arxiv.org/abs/2509.12560 |