In the \emph{dynamic edge coloring} problem, one has to maintain a graph of maximum degree $Δ$ with at most $Δ+c$ colors, given updates to the edges of the graph. An important objective is to minimize the \emph{recourse}, which is the number of edges being recolored. We study this problem on forests, which is a natural yet nontrivial restriction of the problem. We consider the problem in both \emph{incremental} (edges are only inserted) and \emph{fully dynamic} (edges may be deleted) models. In the deterministic setting, we show that the natural greedy algorithm achieves $O(\frac{1}{c + \sqrtΔ})$ amortized recourse in the incremental model, and this is tight up to tie-breaking. In contrast, in a fully dynamic forest, greedy can be forced to have $Ω(\log_Δn)$ amortized recourse. To partially alleviate this limitation of greedy, we show an optimal non-greedy algorithm with $O(1)$ amortized recourse for \emph{rooted} fully dynamic forests and $c = Δ- 2$. In the randomized setting, we give a natural distribution-maintaining algorithm that achieves $Θ(\frac{1}Δ)$ expected amortized recourse in the incremental model and $Θ(\min \{ \fracΔ{c}, \log_Δ n \})$ expected recourse in the dynamic model. These randomized results are optimal for $c=0$.

翻译：在*动态边着色*问题中，给定图边的更新操作，我们需要用至多$Δ+c$种颜色维护一个最大度为$Δ$的图。一个关键目标是最小化*补充操作*(recourse)，即重新着色的边数。我们研究森林上的该问题，这是该问题自然且非平凡的限制。我们考虑*增量式*（仅插入边）和*全动态*（可删除边）两种模型。在确定性设定下，我们证明自然贪心算法在增量模型中实现了$O(\frac{1}{c + \sqrtΔ})$的均摊补充操作次数，且该结果在舍入规则下是紧的。相比之下，在全动态森林中，贪心算法可能被迫达到$Ω(\log_Δ n)$的均摊补充操作次数。为部分缓解贪心算法的这一局限，我们针对*有根*全动态森林且$c = Δ- 2$的情况，给出一个具有$O(1)$均摊补充操作次数的最优非贪心算法。在随机设定下，我们给出一个自然的分布保持算法，在增量模型中实现$Θ(\frac{1}Δ)$的期望均摊补充操作次数，在动态模型中实现$Θ(\min \{ \fracΔ{c}, \log_Δ n \})$的期望补充操作次数。对于$c=0$的情况，这些随机结果是最优的。