  In this paper we introduce the notion of $\Sigma$-colouring of a graph $G$:
For given subsets $\Sigma(v)$ of neighbours of $v$, for every $v\in V(G)$, this
is a proper colouring of the vertices of $G$ such that, in addition, vertices
that appear together in some $\Sigma(v)$ receive different colours. This
concept generalises the notion of colouring the square of graphs and of cyclic
colouring of graphs embedded in a surface. We prove a general result for graphs
embeddable in a fixed surface, which implies asymptotic versions of Wegner's
and Borodin's Conjecture on the planar version of these two colourings. Using a
recent approach of Havet et al., we reduce the problem to edge-colouring of
multigraphs, and then use Kahn's result that the list chromatic index is close
to the fractional chromatic index.
  Our results are based on a strong structural lemma for graphs embeddable in a
fixed surface, which also implies that the size of a clique in the square of a
graph of maximum degree $\Delta$ embeddable in some fixed surface is at most
$\frac32\,\Delta$ plus a constant.
