# Recent Advances in Efficient Dynamic Graph Processing
## 4 Algorithms
### Graph Coloring
#### 4.2.1 Vertex Coloring  
**Definition 7** (Vertex Coloring): _Given a graph \(G\), vertex coloring assigns each vertex in \(G\) a color such that no two incident vertices (connected by an edge) have the same color._  
Vertex coloring seeks to determine an optimal coloring of a graph \(G\), where the chromatic number represents the smallest number of colors required for a valid coloring [140]. This optimization problem is known to be NP-hard for static graphs, as established in [8]. In [141], the computational complexity of optimal graph coloring is established, proving that no polynomial-time algorithm can achieve an \(n^{1-c}\) approximation ratio for any \(\epsilon>0\), unless NP = ZPP. Given this fundamental limitation, researchers have developed numerous approximation approaches for static graph coloring. Ref. [142] presents a systematic survey of existing algorithmic approaches for solving the static graph coloring problem. In addressing the dynamic vertex coloring problem, Ref. [52] proposes two complementary algorithmic frameworks that establish distinct performance trade-offs between recoloring frequency and color minimization. For any \(d>0\), the first algorithm guarantees a proper coloring using \(O(CdN^{1/d})\) colors while recoloring at most \(O(d)\) vertices per update. Conversely, the second algorithm maintains \(O(Cd)\) colors but requires recoloring \(O(dN^{1/d})\) per update, where \(\mathcal{C}\) represents the maximum chromatic number and \(N\) denotes the vertex count. Furthermore, Ref. [52] establishes a lower bound showing that for any 2-colorable graph with \(N\) vertices, maintaining a proper coloring with \(c\) colors(where \(c\geq 2\) is constant) must require at least \(\Omega\big{(}N^{2/c(c-1)}\big{)}\) recolorings per update operation. Ref. [54] studies dynamic graph coloring in the context of bounded-degree graphs. Their work introduces a randomized approach that maintains a proper \((\Delta+1)\)-coloring with an amortized expected update time of \(O(\log\Delta)\), It also presents a deterministic method that achieves a \(\Delta(1+o(1))\)-coloring with an amortized \(O(poly\log\Delta)\) update time, where \(\Delta\) represents the maximum vertex degree. Ref. [49] presents a novel algorithm applicable to general graphs that guarantees a proper \(O(\alpha\log^{2}n)\)-coloring while achieving an amortized time complexity of \(\hat{O}(polylog\alpha)\), where \(\alpha\) denotes the graph's arboricity. Although the graph coloring problem is NP-complete in general, polynomial-time solutions exist for specific dynamic graph classes, including chordal graphs and biconvex bipartite graphs as demonstrated in [143]. From an application-oriented standpoint, dynamic vertex coloring has been investigated in [51], which introduces an algorithmic approach that performs localized updates to the colors of vertices adjacent to modified edges during insertion/deletion operations. Ref. [50] investigates the dynamic graph coloring problem through an edge-centric coloring approach implemented in a distributed graph processing framework. Ref. [53] introduces the most advanced practical solution for sequential dynamic graph coloring. The algorithm proposed in [53] addresses two key requirements: maintaining stable coloring across updates (effectiveness) and limiting computation to affected neighborhoods (efficiency). The solution employs a propagation mechanism indexed by color, which tracks essential color data per vertex, and naturally restricts processing to adjacent vertices of color-changed nodes. **Algorithm analysis. ** We analyze different dynamic vertex coloring algorithms over dynamic graphs. As shown in Table 3, most vertex coloring algorithms are designed to supportfully dynamic graphs. However, research on streaming graphs remains limited and represents an area that requires further exploration and development. For the time complexity, in [54], \(\Delta\) denotes the maximum degree of vertices in the graph.