Gespeichert in:
| Hauptverfasser: | , |
|---|---|
| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2401.02155 |
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Inhaltsangabe:
- 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.