Bounds and Hardness Results for Conflict-free Choosability
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arXiv
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| Natura: | Preprint |
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2024
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| _version_ | 1866909038580596736 |
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| author | Gupta, Shiwali Mathew, Rogers |
| author_facet | Gupta, Shiwali Mathew, Rogers |
| contents | A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other vertex in that hyperedge. The minimum number of colors required for such a coloring is known as the '(partial) conflict-free chromatic number' of $\mathcal{H}$. It is easy to see that the conflict-free chromatic number of a hypergraph is at most its partial conflict-free chromatic number plus one. Conflict-free coloring has also been studied on the open/closed neighborhood hypergraphs of a given graph under the name open/closed neighborhood conflict-free coloring. In this paper, we study partial and full list variants of conflict-free coloring where, for every vertex $v$, we are given a list of admissible colors $L_v$ such that $v$ is allowed to be colored only from $L_v$.
Bhyravarapu, Kalyanasundaram, and Mathew [Journal of Graph Theory, 2021] showed that the closed-neighborhood conflict-free chromatic number of any graph $G$ with maximum degree $Δ$ is at most $O(\ln^2 Δ)$. In this paper, we extend the $O(\ln^2 Δ)$ upper bound to the partial list variant of the closed-neighborhood conflict-free chromatic number. Further, we establish computational complexity results concerning the list open/closed-neighborhood conflict-free chromatic numbers. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2409_12672 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Bounds and Hardness Results for Conflict-free Choosability Gupta, Shiwali Mathew, Rogers Combinatorics Discrete Mathematics 05C15, 05C35, 05D40 G.2.2; G.2.1 A '(partial) conflict-free coloring' of a hypergraph $\mathcal{H}$ is an assignment of colors to (a subset of) the vertex set of $\mathcal{H}$ such that every hyperedge in $\mathcal{H}$ has a vertex whose color is distinct from every other vertex in that hyperedge. The minimum number of colors required for such a coloring is known as the '(partial) conflict-free chromatic number' of $\mathcal{H}$. It is easy to see that the conflict-free chromatic number of a hypergraph is at most its partial conflict-free chromatic number plus one. Conflict-free coloring has also been studied on the open/closed neighborhood hypergraphs of a given graph under the name open/closed neighborhood conflict-free coloring. In this paper, we study partial and full list variants of conflict-free coloring where, for every vertex $v$, we are given a list of admissible colors $L_v$ such that $v$ is allowed to be colored only from $L_v$. Bhyravarapu, Kalyanasundaram, and Mathew [Journal of Graph Theory, 2021] showed that the closed-neighborhood conflict-free chromatic number of any graph $G$ with maximum degree $Δ$ is at most $O(\ln^2 Δ)$. In this paper, we extend the $O(\ln^2 Δ)$ upper bound to the partial list variant of the closed-neighborhood conflict-free chromatic number. Further, we establish computational complexity results concerning the list open/closed-neighborhood conflict-free chromatic numbers. |
| title | Bounds and Hardness Results for Conflict-free Choosability |
| topic | Combinatorics Discrete Mathematics 05C15, 05C35, 05D40 G.2.2; G.2.1 |
| url | https://arxiv.org/abs/2409.12672 |