Fully-dynamic $α+ 2$ Arboricity Decomposition and Implicit Colouring

In the implicit dynamic colouring problem, the task is to maintain a representation of a proper colouring as a dynamic graph is subject to insertions and deletions of edges, while facilitating interspersed queries to the colours of vertices. The goal is to use few colours, while still efficiently handling edge-updates and responding to colour-queries. For an n-vertex dynamic graph of arboricity $\alpha$, we present an algorithm that maintains an implicit vertex colouring with $4\cdot2^\alpha$ colours, in amortised poly-$(\log n)$ update time, and with $O({\alpha} log n)$ worst-case query time. The previous best implicit dynamic colouring algorithm uses $2^{40\alpha}$) colours, and has a more efficient update time of $O(\log³ n)$ and the same query time of $O({\alpha} log n)$ [Henzinger et al'20]. For graphs undergoing arboricity $\alpha$ preserving updates, we give a fully-dynamic $\alpha+2$ arboricity decomposition in poly$(\log n,\alpha)$ time, which matches the number of forests in the best near-linear static algorithm by Blumenstock and Fischer [2020] who obtain $\alpha+2$ forests in near-linear time. Our construction goes via dynamic bounded out-degree orientations, where we present a fully-dynamic explicit, deterministic, worst-case algorithm for $\lfloor (1+\varepsilon)\alpha \rfloor + 2$ bounded out-degree orientation with update time $O(\varepsilon^{-6}\alpha2 \log³ n)$. The state-of-the-art explicit, deterministic, worst-case algorithm for bounded out-degree orientations maintains a $\beta\cdot \alpha + \log{\beta} n$ out-orientation in $O(\beta^{2\alpha2+\beta\alpha\log}{\beta} n)$ time [Kopelowitz et al'13].