Deterministic Dynamic Edge-Colouring

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.