Beyond Vizing Chains: Improved Recourse in Dynamic Edge Coloring

We study the maintenance of a $(\Delta+C)$-edge-coloring ($C\ge 1$) in a fully dynamic graph $G$ with maximum degree $\Delta$. 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 $O\big( \frac{\log n}{\log \frac{\Delta+C}{\Delta-C}}\big)$ for all $C \ge 0.62\Delta$ where $\Delta-C = O(n^{1-\delta})$. Our algorithm is the first deterministic algorithm to establish tight bounds for large palettes, and the first to do so when $\Delta-C=o(\Delta)$. 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 $\alpha$. Previous works [BCPS24, CRV24] achieve $O(\epsilon^{-1}\log n)$ recourse per update with $C\ge (4+\epsilon)\alpha$, and we improve by achieving the same recourse while only requiring $C \ge (2+\epsilon)\alpha - 1$. This result is $\Delta$-adaptive, i.e., it uses $\Delta_t+C$ colors where $\Delta_t$ 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].