🤖 AI Summary
This paper studies the dynamic edge coloring problem: maintaining a deterministic $(1+varepsilon)Delta$-edge coloring in an $n$-vertex graph subject to edge insertions and deletions, where $Delta$ denotes the current maximum degree—thereby breaking the classical $2Delta-1$ bound of greedy static algorithms. We propose the first deterministic dynamic edge coloring algorithm, whose core innovation is a shallow degree-splitter hierarchy that simultaneously achieves low update overhead and strong chromatic guarantees. Via amortized analysis and an exponentially small amortized update time construction, our algorithm achieves an amortized update time of $2^{ ilde{O}(sqrt{log n})}$. When $varepsilon^{-1}$ is subpolynomial, this bound is strictly subpolynomial—marking a substantial improvement over all prior deterministic algorithms.
📝 Abstract
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 $2^{ ilde{O}_{log varepsilon^{-1}}(sqrt{log n})}$ 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 $2Delta-1$ 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.