Improved Dynamic Colouring of Sparse Graphs

Given a dynamic graph subject to edge insertions and deletions, we show how to update an implicit representation of a proper vertex colouring, such that colours of vertices are computable upon query time. We give a deterministic algorithm that uses O(alpha(2)) colours for a dynamic graph of arboricity alpha, and a randomised algorithm that uses O(min{alpha log alpha, alpha log log log n}) colours in the oblivious adversary model. Our deterministic algorithm has update- and query times polynomial in alpha and log n, and our randomised algorithm has amortised update- and query time that with high probability is polynomial in log n with no dependency on the arboricity. Thus, we improve the number of colours exponentially compared to the state-of-the art for implicit colouring, namely from O(2(alpha)) colours, and we approach the theoretical lower bound of Omega(alpha) for this arboricity-parameterised approach. Simultaneously, our randomised algorithm improves the update- and query time to run in time solely polynomial in log n with no dependency on alpha. Our algorithms are fully adaptive to the current value of the dynamic arboricity at query or update time.