A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min\, \{h, {\rm deg}(v)\}$ colours appearing exactly once in its neighbourhood. Let $χ_{\rm pcf}^h(G)$ denote the minimum $k$ such that such a colouring exists. We show that for every fixed $h\ge 1$, every graph $G$ of maximum degree $Δ$ satisfies $χ_{\rm pcf}^h(G) \le hΔ+ \mathcal{O}(\log Δ)$. This expands on the work of Cho et al., and improves a recent result of Liu and Reed in the case $h=1$. We conjecture that for every $h\ge 1$ and every graph $G$ of maximum degree $Δ$ sufficiently large, the bound $χ_{\rm pcf}^h(G) \le hΔ+ 1$ should hold, which would be tight. When the minimum degree $δ$ of $G$ is sufficiently large, namely $δ\ge \max\{100h, 2000\log Δ\}$, we show that this upper bound can be further reduced to $χ_{\rm{pcf}}^h(G) \le Δ+ \mathcal{O}(\sqrt{hΔ})$. This improves a recent bound from Kamyczura and Przybyło when $δ\le \sqrt{hΔ}$.

翻译：图$G$的一个正常$k$着色被称为$h$-无冲突的，如果每个顶点$v$的邻域中至少出现$\min\, \{h, {\rm deg}(v)\}$种颜色恰好一次。令$χ_{\rm pcf}^h(G)$表示存在此类着色的最小$k$值。我们证明，对于每个固定的$h\ge 1$，每个最大度为$Δ$的图$G$满足$χ_{\rm pcf}^h(G) \le hΔ+ \mathcal{O}(\log Δ)$。这扩展了Cho等人的工作，并改进了Liu和Reed在$h=1$情形下的近期结果。我们猜想，对于每个$h\ge 1$以及每个最大度$Δ$足够大的图$G$，界$χ_{\rm pcf}^h(G) \le hΔ+ 1$应当成立，且该界是紧的。当图$G$的最小度$δ$足够大，即满足$δ\ge \max\{100h, 2000\log Δ\}$时，我们证明该上界可进一步改进为$χ_{\rm{pcf}}^h(G) \le Δ+ \mathcal{O}(\sqrt{hΔ})$。这改进了Kamyczura和Przybyło在$δ\le \sqrt{hΔ}$情形下的近期界。