Caro–Petru1evski–skrekovski's conflict-free chromatic conjecture

Let $G$ be a connected graph with maximum degree $\Delta$, and let $\chi_{\rm pcf}(G)$ be its conflict-free chromatic number, the least number of colours in a proper vertex colouring such that every vertex has a neighbour whose colour occurs exactly once in its neighbourhood.

Caro–Petru1evski–skrekovski's conjecture. For every connected graph $G$ of maximum degree $\Delta\geq 3$,

$$\chi_{\rm pcf}(G)\leq \Delta+1.$$

This conjecture strengthens the known general upper bound $\chi_{\rm pcf}(G)\leq\left\lfloor 5\Delta/2\right\rfloor$ and is the central conflict-free colouring problem addressed asymptotically in the paper. The text indicates that it is proved when the minimum degree is sufficiently large relative to $\log\Delta$, while the unrestricted case remains open.