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 $\operatorname{poly}(\log n)$ amortised 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 $\max\{deg(u), deg(v)\} + 2\alpha$ colours in $O(m\log n)$ time. This 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 $\Delta$.

翻译：我们研究边着色问题，并给出以图的树状度α参数化颜色数的高效算法。在支持插入和删除操作的动态图中，我们提出一种确定性算法，可在poly(log n)均摊时间内更新得到Δ+O(α)边着色。该算法完全自适应于当前最大度和树状度的取值。在全动态场景下，现有最优边着色算法要么是使用(1+ε)Δ种颜色的随机算法（每次更新需poly(log n, ε^{-1})时间），要么是朴素贪心算法（确定性2Δ-1边着色，更新时间为log(Δ)）。与(1+ε)Δ算法相比，我们的算法是确定性的且渐进更快，当α相比Δ足够小时，甚至使用更少的颜色。特别地，这是首个针对动态森林和动态平面图、具有多对数更新时间的Δ+O(1)边着色算法。此外在静态场景中，我们证明可在O(m log n)时间内找到使用max{deg(u), deg(v)}+2α种颜色的正确边着色。该时间复杂度与计算2Δ-1着色的贪心算法相当，并在α相比Δ足够小时改进了颜色数量。