Asymptotically Optimal Proper Conflict-Free Colouring

TL;DR

This work resolves the asymptotic behaviour of proper conflict-free colourings for graphs with maximum degree $\Delta$ by proving a $(1+o(1))\Delta$-style bound. The authors develop a multi-stage probabilistic construction that splits the vertex set into a low-degree part $L$ and a high-degree part $H$, colours $G[L]$ greedily to achieve local uniqueness, and then refines the colouring of $H$ through auxiliary graphs, a subset $Z$, a partition of a refined high-degree subset, and disjoint palettes across parts. Central to the approach are two key lemmas (one constructing a suitable $Y\subseteq H$ and another providing a compatible partition of $H'$) proven via Lovász Local Lemma, Chernoff-type bounds, and Azuma/McDiarmid concentration, yielding an overall bound of at most $\Delta+O(\Delta^{2/3}\log\Delta)$ colours. The result advances the long-standing conjecture by achieving the asymptotic optimality and highlights a powerful combination of decomposition, probabilistic colouring, and graph augmentations for conflict-free colouring problems in general graphs.

Abstract

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.

Theorems & Definitions (20)

• Conjecture 1.1: cps
• Theorem 1.2
• Theorem 2.1: Chernoff Bound (see Page 43 in mr_book)
• Theorem 2.2: Lovász Local Lemma el
• Theorem 2.3: Azuma's Inequality a
• Theorem 2.4: mc
• Lemma 2.5
• Corollary 2.6
• Lemma 3.1
• proof