  We study the edge-colouring problem, and give efficient algorithms where the
number of colours is parameterised by the graph's arboricity, $\alpha$. In a
dynamic graph, subject to insertions and deletions, we give a deterministic
algorithm that updates a proper $\Delta + O(\alpha)$ edge~colouring in
$\operatorname{poly}(\log n)$ amortized time. Our algorithm is fully adaptive
to the current value of the maximum degree and arboricity.
  In this fully-dynamic setting, the state-of-the-art edge-colouring algorithms
are either a randomised algorithm using $(1 + \varepsilon)\Delta$ colours in
$\operatorname{poly}(\log n, \epsilon^{-1})$ time per update, or the naive
greedy algorithm which is a deterministic $2\Delta -1$ edge colouring with
$\log(\Delta)$ update time.
  Compared to the $(1+\varepsilon)\Delta$ algorithm, our algorithm is
deterministic and asymptotically faster, and when $\alpha$ is sufficiently
small compared to $\Delta$, it even uses fewer colours. In particular, ours is
the first $\Delta+O(1)$ edge-colouring algorithm for dynamic forests, and
dynamic planar graphs, with polylogarithmic update time.
  Additionally, in the static setting, we show that we can find a proper edge
colouring with $\Delta + 2\alpha$ colours in $O(m\log n)$ time. Moreover, the
colouring returned by our algorithm has the following local property: every
edge $uv$ is coloured with a colour in $\{1, \max\{deg(u), deg(v)\} +
2\alpha\}$. The time bound matches that of the greedy algorithm that computes a
$2\Delta-1$ colouring of the graph's edges, and improves the number of colours
when $\alpha$ is sufficiently small compared to $\Delta$.
