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中维护一个（Δ+C）边着色问题（C≥1），重点关注最小化每次边更新时重新着色边数的资源消耗。我们提出一种基于称为**移位树**对象的新技术。该对象追踪G的多种可能重着色方案，使我们能在多项式时间内以低资源消耗维护合法着色。我们沿边路径移动颜色，但不同于许多其他算法，我们不使用**扇**和**双色交替路径**。通过将移位树与附加技术结合，我们得到一种算法，对于所有满足Δ-C=O(n^{1-δ})的C≥0.62Δ情形，实现紧密的资源消耗上界O( (log n)/(log((Δ+C)/(Δ-C))) )。这是首个为大调色板建立紧密界，且在Δ-C=o(Δ)情形下达成该界的确定性算法。该结果解决了大调色板场景下资源消耗的理论复杂性。此外，我们认为将可能的移位视为树结构可衍生出类似的树基技术，这些技术可扩展至更小的C值，并实现更优的更新时间。第二个应用涉及低树状度α的图。先前工作[BCPS24, CRV24]在C≥(4+ε)α时每次更新达到O(ε^{-1}log n)的资源消耗，而我们在仅需C≥(2+ε)α-1的条件下实现相同资源消耗。此结果具有Δ自适应性，即使用Δ_t+C种颜色（其中Δ_t为当前最大度）。为探究技术局限性及基于移位的算法的一般特性，我们证明了仅沿路径移动颜色的算法与采用Nibbling方法[BGW21, BCPS24]的广义算法之间在可实现的资源消耗上存在分离。