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, $α$. In a dynamic graph, subject to insertions and deletions, we give a deterministic algorithm that updates a proper $Δ+ O(α)$ 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 $(1 + \varepsilon)Δ$ colours in $\operatorname{poly}(\log n, ε^{-1})$ time per update, or the naive greedy algorithm which is a deterministic $2Δ-1$ edge colouring with $\log(Δ)$ update time. Compared to the $(1+\varepsilon)Δ$ algorithm, our algorithm is deterministic and asymptotically faster, and when $α$ is sufficiently small compared to $Δ$, it even uses fewer colours. In particular, ours is the first $Δ+O(1)$ 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 $Δ+ 2α$ 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 $\{1, \max\{deg(u), deg(v)\} + 2α\}$. The time bound matches that of the greedy algorithm that computes a $2Δ-1$ colouring of the graph's edges, and improves the number of colours when $α$ is sufficiently small compared to $Δ$.