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+ε)Δ-edge-colouring of G without the use of randomisation. More specifically, we show a deterministic dynamic algorithm with an amortised update time of 2^Õ_log ε⁻¹(√log n) using (1+ε)Δ colours. If ε⁻¹ ∈ 2^O(log0.49 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 2Δ-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 nᵒ⁽¹⁾. We believe that this algorithm might be of independent interest.