Asymptotically Optimal Proper Conflict-Free Colouring

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 $\mathrm{\Delta }\ge 3$ has a proper conflict-free colouring with at most $\mathrm{\Delta }+1$ colours. This conjecture holds for $\mathrm{\Delta }=3$ and remains open for $\mathrm{\Delta }\ge 4$. In this paper we prove that this conjecture holds asymptotically; namely, every graph with maximum degree $\mathrm{\Delta }$ has a proper conflict-free colouring with $(1+o(1))\mathrm{\Delta }$ colours.