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 $\min, {h, {\rm deg}(v)}$ 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 $δ\le \sqrt{hΔ}$.

💡 Research Summary

The paper studies proper h‑conflict‑free (h‑PCF) colourings of graphs, a natural generalisation of the conflict‑free colourings introduced by Fabrici et al. In an h‑PCF colouring each vertex v must have at least min {h, deg(v)} colours that appear exactly once among its neighbours; such colours are called solitary and the corresponding neighbours are witnesses. The authors denote by χₚ𝚌𝚏^h(G) the smallest number of colours that admits an h‑PCF colouring of G.

The main contributions are two asymptotic upper bounds that improve on all previously known results. First, for any fixed integer h ≥ 1 and any connected graph G with maximum degree Δ, they prove
 χₚ𝚌𝚏^h(G) ≤ h Δ + O(h log Δ).
The hidden constant in the O‑term is made explicit in Theorem 24. This result subsumes the recent bound of Liu and Reed for the case h = 1 (which gave Δ + O(Δ^{2/3} log Δ)) and improves the earlier bound of Cho et al. (h+1)Δ − 1. The proof proceeds in two stages. In the first stage a modified greedy algorithm colours the vertices using h Δ + 1 colours while guaranteeing that every vertex of “small degree” (degree at most h h+1 Δ) already has h solitary colours, and every vertex of larger degree has h − 1 solitary colours. The algorithm works by ordering the vertices so that low‑degree vertices are coloured last, and by reserving a set W(v) of the first min{h,deg(v)} (or h − 1) neighbours in the order as witnesses. In the second stage a Rödl‑Nibble‑type random recolouring is applied to a carefully chosen fraction of vertices. Using Chernoff bounds, a coupling argument (Lemma 15), and the symmetric Lovász Local Lemma (in both its standard and lopsided forms), the authors show that with positive probability each high‑degree vertex gains one additional solitary colour without destroying the witnesses already created. Thus the total number of colours never exceeds h Δ + O(h log Δ).

The second major result concerns graphs whose minimum degree δ is not too small. If δ ≥ max{100 h, 2000 log Δ}, the authors prove
 χₚ𝚌𝚏^h(G) ≤ Δ + O(√{h Δ}).
This improves the recent bound of Kamyczura and Przybyło, which gave Δ + O(Δ log Δ/δ) and is weaker when δ = Θ(√{h Δ}). The same two‑stage strategy is employed, but the initial greedy phase now treats a larger set of vertices as “small degree”, thereby providing more witnesses up front. The random recolouring phase again uses a nibble, but the analysis is refined to exploit the higher minimum degree, leading to the √{h Δ} term.

The paper also discusses lower bounds. Proposition 6 shows that for any fixed h there exist graphs with arbitrarily large Δ for which χₚ𝚌𝚏^h(G) ≥ h Δ + 1, establishing that the leading term h Δ in the upper bounds cannot be improved. Consequently the authors conjecture (Conjecture 7) that for every fixed h and sufficiently large Δ the optimal bound is χₚ𝚌𝚏^h(G) ≤ h Δ + 1, which would be tight. They also relate h‑PCF colourings to h‑odd colourings, noting that χₒ^h(G) ≤ χₚ𝚌𝚏^h(G) and that the same lower‑bound construction applies, reinforcing the conjecture’s plausibility.

Finally, the paper collects the probabilistic tools used throughout: Chernoff bounds (Lemma 12), simple tail estimates for binomial variables (Lemma 14), a coupling lemma for dependent binary variables (Lemma 15), and both the standard and lopsided Lovász Local Lemma (Lemmas 16 and 17). These are applied to control the random recolouring step and to guarantee that the set of “bad events” (vertices lacking enough witnesses) has a non‑zero probability of being avoided.

In summary, the authors deliver a substantial improvement on the asymptotic behaviour of h‑conflict‑free colourings: the additive error term is reduced from a polynomial in Δ to a logarithmic term for arbitrary h, and to a √{h Δ} term when the graph has sufficiently large minimum degree. The work combines refined greedy constructions with sophisticated probabilistic nibble techniques, and it proposes a natural and likely optimal conjecture for the exact leading constant.