Wegner's conjecture on the 2-distance chromatic number of planar graphs

Let $G$ be a planar graph with maximum degree $\Delta$, and let $\chi_2(G)$ denote the minimum number of colors in a coloring in which vertices at distance at most $2$ receive distinct colors. Wegner's conjecture.

$$\chi_2(G)\leq 7 \quad\text{if } \Delta=3,$$ $$\chi_2(G)\leq \Delta+5 \quad\text{if } 4\leq\Delta\leq 7,$$

and

$$\chi_2(G)\leq \left\lfloor\frac{3\Delta}{2}\right\rfloor+1 \quad\text{if } \Delta\geq 8.$$

The conjecture remains largely open: the source states that the case $\Delta=3$ was proved by Thomassen, while various weaker upper bounds are known for larger maximum degrees.