Single conflict coloring, adaptable choosability and separation choosability
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916954695008256 |
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| author | Casselgren, Carl Johan Eriksson, Kalle |
| author_facet | Casselgren, Carl Johan Eriksson, Kalle |
| contents | We study relations between three interrelated notions of graph (list) coloring: single conflict coloring, adapted list coloring and choosability with separation (with $1$ overlapping color between lists of adjacent vertices), and their respective invariants single conflict chromatic number $χ_{\nleftrightarrow}$, adaptable choosability $ch_{ad}$ and separation choosability $ch_{sep}$. We investigate graphs with small values of these invariants, and construct explicit families of graphs $G$ with $χ_{\nleftrightarrow}(G) = ch_{ad}(G) > ch_{sep}(G)$, as well as where all three invariants are equal. Furthermore, we consider planar graphs and investigate for which triples $(a,b,c)$, there is a planar graph $G$ with $(ch_{sep}(G), ch_{ad}(G), χ_{\nleftrightarrow}(G)) = (a,b,c)$. Throughout the paper we pose many questions on these graph coloring parameters, and discuss connections to related coloring invariants such as adapted coloring. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_13913 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Single conflict coloring, adaptable choosability and separation choosability Casselgren, Carl Johan Eriksson, Kalle Combinatorics We study relations between three interrelated notions of graph (list) coloring: single conflict coloring, adapted list coloring and choosability with separation (with $1$ overlapping color between lists of adjacent vertices), and their respective invariants single conflict chromatic number $χ_{\nleftrightarrow}$, adaptable choosability $ch_{ad}$ and separation choosability $ch_{sep}$. We investigate graphs with small values of these invariants, and construct explicit families of graphs $G$ with $χ_{\nleftrightarrow}(G) = ch_{ad}(G) > ch_{sep}(G)$, as well as where all three invariants are equal. Furthermore, we consider planar graphs and investigate for which triples $(a,b,c)$, there is a planar graph $G$ with $(ch_{sep}(G), ch_{ad}(G), χ_{\nleftrightarrow}(G)) = (a,b,c)$. Throughout the paper we pose many questions on these graph coloring parameters, and discuss connections to related coloring invariants such as adapted coloring. |
| title | Single conflict coloring, adaptable choosability and separation choosability |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2509.13913 |