Proper conflict-free choosability of planar graphs
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909976969084928 |
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| author | Wang, Yuting Zhang, Xin |
| author_facet | Wang, Yuting Zhang, Xin |
| contents | A proper conflict-free coloring of a graph is a proper vertex coloring wherein each non-isolated vertex's open neighborhood contains at least one color appearing exactly once. For a non-negative integer $k$, a graph $G$ is said to be proper conflict-free (degree+$k$)-choosable if given any list assignment $L$ for $G$ where $|L(v)| = d(v) + k$ holds for every vertex $v \in V(G)$, there exists a proper conflict-free coloring $ϕ$ of $G$ such that $ϕ(v) \in L(v)$ for all $v \in V(G)$. Recently, Kashima, Škrekovski, and Xu proposed two related conjectures on proper conflict-free choosability: the first asserts the existence of an absolute constant $k$ such that every graph is proper conflict-free (degree+$k$)-choosable, while the second strengthens this claim by restricting to connected graphs other than the cycle of length 5 and reducing the constant to $k=2$. In this paper, we confirm the second conjecture for three graph classes: $K_4$-minor-free graphs with maximum degree at most 4, outer-1-planar graphs with maximum degree at most 4, and planar graphs with girth at least 12; we also confirm the first conjecture for these same graph classes, in addition to all outer-1-planar graphs (without degree constraints). Moreover, we prove that planar graphs with girth at least 12 and outer-1-planar graphs are proper conflict-free $6$-choosable. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_22805 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Proper conflict-free choosability of planar graphs Wang, Yuting Zhang, Xin Combinatorics Discrete Mathematics A proper conflict-free coloring of a graph is a proper vertex coloring wherein each non-isolated vertex's open neighborhood contains at least one color appearing exactly once. For a non-negative integer $k$, a graph $G$ is said to be proper conflict-free (degree+$k$)-choosable if given any list assignment $L$ for $G$ where $|L(v)| = d(v) + k$ holds for every vertex $v \in V(G)$, there exists a proper conflict-free coloring $ϕ$ of $G$ such that $ϕ(v) \in L(v)$ for all $v \in V(G)$. Recently, Kashima, Škrekovski, and Xu proposed two related conjectures on proper conflict-free choosability: the first asserts the existence of an absolute constant $k$ such that every graph is proper conflict-free (degree+$k$)-choosable, while the second strengthens this claim by restricting to connected graphs other than the cycle of length 5 and reducing the constant to $k=2$. In this paper, we confirm the second conjecture for three graph classes: $K_4$-minor-free graphs with maximum degree at most 4, outer-1-planar graphs with maximum degree at most 4, and planar graphs with girth at least 12; we also confirm the first conjecture for these same graph classes, in addition to all outer-1-planar graphs (without degree constraints). Moreover, we prove that planar graphs with girth at least 12 and outer-1-planar graphs are proper conflict-free $6$-choosable. |
| title | Proper conflict-free choosability of planar graphs |
| topic | Combinatorics Discrete Mathematics |
| url | https://arxiv.org/abs/2512.22805 |