New bounds for proper $h$-conflict-free colourings
Quentin Chuet, Tianjiao Dai, Qiancheng Ouyang, François Pirot
TL;DR
The paper studies proper $h$-conflict-free colourings in graphs by introducing $\chi_{\rm pcf}^h(G)$ and proving upper bounds in terms of the maximum degree $\Delta$. It develops a two-phase colouring strategy—a greedy precolouring to secure witnesses for small-degree vertices, followed by a Rödl nibble recolouring to create additional witnesses for large-degree vertices—and leverages probabilistic tools including Chernoff bounds and the Lovász Local Lemma. The main results show $\chi_{\rm pcf}^h(G) \le h\Delta + \mathcal{O}(\log \Delta)$ for fixed $h$, and, under a sufficiently large minimum degree $\delta$, $\chi_{\rm pcf}^h(G) \le \Delta + \mathcal{O}(\sqrt{h\Delta})$, with explicit constants and regimes. These improvements extend prior bounds, connect to odd colourings, and illuminate how minimum degree affects the second-order term in the bound, clarifying a path toward the conjectured $h\Delta+1$ threshold.
Abstract
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 $δ\ge \max\{100h, 3000\log Δ\}$, 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 $δ\le \sqrt{hΔ}$.