We study the maintenance of a $(Δ+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $Δ$. We focus on minimizing \emph{recourse} which equals the number of recolored edges per edge updates. We present a new technique based on an object which we call a \emph{shift-tree}. This object tracks multiple possible recolorings of $G$ and enables us to maintain a proper coloring with small recourse in polynomial time. We shift colors over a path of edges, but unlike many other algorithms, we do not use \emph{fans} and \emph{alternating bicolored paths}. We combine the shift-tree with additional techniques to obtain an algorithm with a \emph{tight} recourse of $O\big( \frac{\log n}{\log \frac{Δ+C}{Δ-C}}\big)$ for all $C \ge 0.62Δ$ where $Δ-C = O(n^{1-δ})$. Our algorithm is the first deterministic algorithm to establish tight bounds for large palettes, and the first to do so when $Δ-C=o(Δ)$. This result settles the theoretical complexity of the recourse for large palettes. Furthermore, we believe that viewing the possible shifts as a tree can lead to similar tree-based techniques that extend to lower values of $C$, and to improved update times. A second application is to graphs with low arboricity $α$. Previous works [BCPS24, CRV24] achieve $O(ε^{-1}\log n)$ recourse per update with $C\ge (4+ε)α$, and we improve by achieving the same recourse while only requiring $C \ge (2+ε)α- 1$. This result is $Δ$-adaptive, i.e., it uses $Δ_t+C$ colors where $Δ_t$ is the current maximum degree. Trying to understand the limitations of our technique, and shift-based algorithms in general, we show a separation between the recourse achievable by algorithms that only shift colors along a path, and more general algorithms such as ones using the Nibbling Method [BGW21, BCPS24].

翻译：我们研究在最大度为Δ的全动态图G中维护(Δ+C)-边着色（C≥1）的问题。我们重点关注最小化资源消耗，即每次边更新时重新着色的边数。我们提出一种基于称为移位树的新技术。该对象追踪G的多种可能重新着色方案，使我们能够在多项式时间内以较小的资源消耗维护合法着色。我们沿着边路径移动颜色，但与许多其他算法不同，我们不使用扇结构和交替双色路径。我们将移位树与其他技术结合，得到对所有C≥0.62Δ且Δ-C=O(n^(1-δ))的情况具有紧界资源消耗O(log n / log((Δ+C)/(Δ-C)))的算法。我们的算法是首个为大调色板建立紧界的确定性算法，也是首个在Δ-C=o(Δ)时实现该目标的算法。该结果解决了大调色板资源消耗的理论复杂度问题。此外，我们认为将可能的移位视为树结构可以推广到更小的C值，并改进更新时间。第二个应用针对低树密度α的图。先前研究[BCPS24, CRV24]在C≥(4+ε)α时实现每次更新O(ε^(-1)log n)的资源消耗，我们通过仅需C≥(2+ε)α-1即可实现相同资源消耗而作出改进。该结果是Δ自适应的，即使用Δ_t+C种颜色，其中Δ_t是当前最大度。为探究我们技术及一般移位算法的局限性，我们证明了仅沿路径移动颜色的算法与使用Nibbling方法[BGW21, BCPS24]等更通用算法所能达到的资源消耗之间存在分离。