Asymptotically Optimal Proper Conflict-Free Colouring
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915266012643328 |
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| author | Liu, Chun-Hung Reed, Bruce |
| author_facet | Liu, Chun-Hung Reed, Bruce |
| contents | A proper conflict-free colouring of a graph is a colouring of the vertices such that any two adjacent vertices receive different colours, and for every non-isolated vertex $v$, some colour appears exactly once on the neighbourhood of $v$. Caro, Petruševski and Škrekovski conjectured that every connected graph with maximum degree $Δ\geq 3$ has a proper conflict-free colouring with at most $Δ+1$ colours. This conjecture holds for $Δ=3$ and remains open for $Δ\geq 4$. In this paper we prove that this conjecture holds asymptotically; namely, every graph with maximum degree $Δ$ has a proper conflict-free colouring with $(1+o(1))Δ$ colours. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2401_02155 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asymptotically Optimal Proper Conflict-Free Colouring Liu, Chun-Hung Reed, Bruce Combinatorics A proper conflict-free colouring of a graph is a colouring of the vertices such that any two adjacent vertices receive different colours, and for every non-isolated vertex $v$, some colour appears exactly once on the neighbourhood of $v$. Caro, Petruševski and Škrekovski conjectured that every connected graph with maximum degree $Δ\geq 3$ has a proper conflict-free colouring with at most $Δ+1$ colours. This conjecture holds for $Δ=3$ and remains open for $Δ\geq 4$. In this paper we prove that this conjecture holds asymptotically; namely, every graph with maximum degree $Δ$ has a proper conflict-free colouring with $(1+o(1))Δ$ colours. |
| title | Asymptotically Optimal Proper Conflict-Free Colouring |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2401.02155 |