Sparsity-Parameterised Dynamic Edge Colouring

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 $\Delta$.