Given a dynamic graph $G$ with $n$ vertices and $m$ edges subject to insertion an deletions of edges, we show how to maintain a $(1+\varepsilon)\Delta$-edge-colouring of $G$ without the use of randomisation. More specifically, we show a deterministic dynamic algorithm with an amortised update time of $2^{\tilde{O}_{\log \varepsilon^{-1}}(\sqrt{\log n})}$ using $(1+\varepsilon)\Delta$ colours. If $\varepsilon^{-1} \in 2^{O(\log^{0.49} n)}$, then our update time is sub-polynomial in $n$. While there exists randomised algorithms maintaining colourings with the same number of colours [Christiansen STOC'23, Duan, He, Zhang SODA'19, Bhattacarya, Costa, Panski, Solomon SODA'24] in polylogarithmic and even constant update time, this is the first deterministic algorithm to go below the greedy threshold of $2\Delta-1$ colours for all input graphs. On the way to our main result, we show how to dynamically maintain a shallow hierarchy of degree-splitters with both recourse and update time in $n^{o(1)}$. We believe that this algorithm might be of independent interest.

翻译：给定一个动态图$G$，包含$n$个顶点和$m$条边，且支持边的插入与删除操作，我们展示如何在不使用随机化的情况下维护$G$的一个$(1+\varepsilon)\Delta$-边着色。具体而言，我们提出一种确定性动态算法，其均摊更新时间为$2^{\tilde{O}_{\log \varepsilon^{-1}}(\sqrt{\log n})}$，并使用$(1+\varepsilon)\Delta$种颜色。若$\varepsilon^{-1} \in 2^{O(\log^{0.49} n)}$，则更新时间为$n$的次多项式。尽管存在随机化算法能在多对数甚至常数更新时间内维持相同颜色数量的着色[Christiansen STOC'23, Duan, He, Zhang SODA'19, Bhattacarya, Costa, Panski, Solomon SODA'24]，但这是首个对所有输入图均能突破贪心下界$2\Delta-1$种颜色的确定性算法。在实现主要结果的过程中，我们展示了如何动态维护一个浅层的度数分割器层级结构，其重算次数和更新时间均为$n^{o(1)}$。我们相信该算法可能具有独立研究价值。