  A graph $G$ has maximal local edge-connectivity $k$ if the maximum number of
edge-disjoint paths between every pair of distinct vertices $x$ and $y$ is at
most $k$. We prove Brooks-type theorems for $k$-connected graphs with maximal
local edge-connectivity $k$, and for any graph with maximal local
edge-connectivity 3. We also consider several related graph classes defined by
constraints on connectivity. In particular, we show that there is a
polynomial-time algorithm that, given a 3-connected graph $G$ with maximal
local connectivity 3, outputs an optimal colouring for $G$. On the other hand,
we prove, for $k \ge 3$, that $k$-colourability is NP-complete when restricted
to minimally $k$-connected graphs, and 3-colourability is NP-complete when
restricted to $(k-1)$-connected graphs with maximal local connectivity $k$.
Finally, we consider a parameterization of $k$-colourability based on the
number of vertices of degree at least $k+1$, and prove that, even when $k$ is
part of the input, the corresponding parameterized problem is FPT.
