This article studies the fundamental problem of graph coloring in fully dynamic graphs. Since the problem of computing an optimal coloring, or even approximating it to within n 1-ε for any ε > 0, is NP-hard in static graphs, there is no hope to achieve any meaningful computational results for general graphs in the dynamic setting. It is therefore only natural to consider the combinatorial aspects of dynamic coloring or alternatively, study restricted families of graphs. Toward understanding the combinatorial aspects of this problem, one may assume a black-box access to a static algorithm for C -coloring any subgraph of the dynamic graph, and investigate the trade-off between the number of colors and the number of recolorings per update step. Optimizing the number of recolorings, sometimes referred to as the recourse bound, is important for various practical applications. In WADS ’17, Barba et al. devised two complementary algorithms: for any β > 0, the first (respectively, second) maintains an O(Cβn 1/β ) (respectively, O(Cβ) -coloring while recoloring O(β) (respectively, O(β n 1/β )) vertices per update. Barba et al. also showed that the second trade-off appears to exhibit the right behavior, at least for β = O(1): any algorithm that maintains a C -coloring of an n -vertex dynamic forest must recolor Ω (n 2 C(C-1)) vertices per update, for any constant C ≥ 2. Our contribution is twofold: • We devise a new algorithm for general graphs that improves significantly upon the first trade-off in a wide range of parameters: for any β > 0, we get a Ô (Cβlog 2 n)-coloring with O(β) recolorings per update, where the Ô notation suppresses polyloglog(n) factors. In particular, for β = O(1), we get constant recolorings with polylog(n) colors; not only is this an exponential improvement over the previous bound but also it unveils a rather surprising phenomenon: the trade-off between the number of colors and recolorings is highly non-symmetric. • For uniformly sparse graphs, we use low out-degree orientations to strengthen the preceding result by bounding the update time of the algorithm rather than the number of recolorings. Then, we further improve this result by introducing a new data structure that refines bounded out-degree edge orientations and is of independent interest. From this data structure, we get a deterministic algorithm for graphs of arboricity ɑ that maintains an O(ɑ log 2 n)-coloring in amortized O(1) time.

中文翻译：

---

改进的动态图着色

本文研究了完全动态图中的图着色的基本问题。由于计算最佳着色的问题，甚至将其逼近到 n1-ε对于任何 ε > 0，在静态图中是 NP-hard，对于动态设置中的一般图，没有希望获得任何有意义的计算结果。因此，很自然地考虑动态着色的组合方面，或者研究受限制的图族。为了理解这个问题的组合方面，可以假设对静态算法的黑盒访问C- 为动态图的任何子图着色，并研究颜色数量和每个更新步骤的重新着色次数之间的权衡。优化重新着色的数量，有时称为资源限制，对于各种实际应用很重要。在 WADS '17 中，Barba 等人。设计了两个互补的算法：对于任何 β > 0，第一个（分别是第二个）保持一个O(Cβn1/β)（分别，O(Cβ)-在重新着色时着色 O(β)（分别为 O(βn 1/β)) 每次更新的顶点。巴巴等人。还表明，第二个权衡似乎表现出正确的行为，至少对于 β = O(1)：任何保持C- 着色n-顶点动态森林必须重新着色Ω（n2对于任何常数 C ≥ 2，每次更新 C(C-1)) 个顶点。我们的贡献是双重的： • 我们为一般图设计了一种新算法，该算法在广泛参数的第一次权衡中显着改进：对于任何β > 0，我们得到一个 Ô (Cβlog2n)-每次更新使用 O(β) 重新着色进行着色，其中 Ô 符号抑制多对数对数（n）因素。特别是，对于 β = O(1)，我们得到不断的重新着色多对数（n）颜色; 这不仅是对先前界限的指数改进，而且还揭示了一个相当令人惊讶的现象：颜色数量和重新着色之间的权衡是高度不对称的。• 对于均匀稀疏图，我们使用低出度方向通过限制算法的更新时间而不是重新着色的次数来加强前面的结果。然后，我们通过引入一种新的数据结构来进一步改进这一结果，该数据结构可以细化有界的出度边缘方向并且具有独立的兴趣。从这个数据结构中，我们得到了一个保持 O(ɑ log2n)-在分期 O(1) 时间内着色。