New bounds for proper $h$-conflict-free colourings

A proper $k$-colouring of a graph $G$ is called $h$-conflict-free if every vertex $v$ has at least $\min \{h,\mathrm{d}\mathrm{e}\mathrm{g}(v)\}$ colours appearing exactly once in its neighbourhood. Let ${\chi }_{\mathrm{p}\mathrm{c}\mathrm{f}}^h(G)$ 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 $\mathrm{\Delta }$ satisfies ${\chi }_{\mathrm{p}\mathrm{c}\mathrm{f}}^h(G)\le h\mathrm{\Delta }+\mathcal{O}(\log \mathrm{\Delta })$. 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 $\mathrm{\Delta }$ sufficiently large, the bound ${\chi }_{\mathrm{p}\mathrm{c}\mathrm{f}}^h(G)\le h\mathrm{\Delta }+1$ should hold, which would be tight. When the minimum degree $\delta$ of $G$ is sufficiently large, namely $\delta \ge \max \{100h,2000\log \mathrm{\Delta }\}$, we show that this upper bound can be further reduced to ${\chi }_{\mathrm{p}\mathrm{c}\mathrm{f}}^h(G)\le \mathrm{\Delta }+\mathcal{O}(\sqrt{h\mathrm{\Delta }})$. This improves a recent bound from Kamyczura and Przybyło when $\delta \le \sqrt{h\mathrm{\Delta }}$.