  In 1977, Wegner conjectured that the chromatic number of the square of every
planar graph $G$ with maximum degree $\Delta\ge8$ is at most
$\bigl\lfloor\frac32\Delta\bigr\rfloor+1$. We show that it is at most $\frac32
\Delta (1+o(1))$ (where the $o(1)$ is as $\Delta\to+\infty$), and indeed that
this is true for the list chromatic number and for more general classes of
graphs.

Combinatorics