# Recent Advances in Efficient Dynamic Graph Processing
## 4 Algorithms
### Graph Coloring
#### 4.2.2 Edge Coloring  
**Definition 8** (Edge Coloring).: _Edge coloring of a graph is an assignment of colors to the edges of the graph so that no two incident edges have the same color._  
Ref. [56] introduces the OHEColoring algorithm, an ordering heuristic-based static edge coloring approach with \(O(m\cdot d_{max})\) time complexity (where \(m\) represents edge count and \(d_{max}\) denotes maximum degree). For dynamic maintenance, the work develops: (1) a propagation mechanism that handles edge recoloring triggered by graph updates, and (2) an edge-priority recoloring algorithm that guarantees single recoloring per edge to optimize computational efficiency.  
For dynamic graphs undergoing continuous edge modifications, Refs. [54; 55] present efficient solutions to maintain proper edge coloring. Their approaches ensure a proper coloring using at most \((2\Delta-1)\) colors while requiring only \(O(\log\Delta)\) time complexity per update operation, where \(\Delta\) represents the maximum vertex degree. Ref. [57] demonstrates the maintenance of \((1+\epsilon)\Delta\)-coloring with an amortized update time of \(O(\log^{8}n/\epsilon^{4})\) for graphs satisfying \(\Delta\geq\Omega(\log^{2}n/\epsilon^{2})\). This result is later improved by [59], which achieves the same coloring quality with an update time of \(O(\log^{4}(1/\epsilon)/\epsilon^{9})\), under the relaxed condition \(\Delta\geq(\log n/\epsilon)^{\Theta((1/\epsilon)\log(1/\epsilon))}\). Ref. [60] establishes an efficient algorithm for maintaining a \((\Delta+O(\alpha))\)-edge coloring in dynamic graphs, achieving a constant amortized update time \(\tilde{O}(1)\), where \(\alpha\) denotes the graph's arboricity, \(\Delta\) represents its maximum degree, and \(\epsilon\) is a given parameter.  
Recent advances in \((1+\epsilon)\Delta\)-coloring algorithms demonstrate significant progress: Ref. [57] achieves an update time of \(O(\log^{8}n/\epsilon^{4})\) when \(\Delta=\Omega(\frac{\log^{2}n}{\epsilon^{2}})\), while Ref. [58] removes the degree restriction with \(O(\log^{9}n\log^{6}\Delta/\epsilon^{6})\) complexity. The algorithm in [59] further reduces the update time to \(2^{\tilde{O}\log\epsilon^{-1}(\sqrt{\log n})}\) under certain parameter conditions on \(\Delta\) and \(\epsilon\).  
**Algorithm analysis.** We compare different edge coloring algorithms over dynamic graphs. Most edge coloring algorithms support incremental/decremental dynamic graphs. For the time complexity of existing works, the worst-time complexity of most algorithms depends on \(\Delta\), such as [54; 55] where \(\Delta\) is the maximum degree. Refs. [57; 58; 59; 61] all studies \((1+\epsilon)\Delta\)-coloring problem. The worst-time complexity of [57; 59; 61] is bounded by a polynomial in \(\epsilon\) and \(n\), where \(\epsilon>0\) is an arbitrarily small constant and \(n\) is the number of vertices. However, there is a noticeable gap in research concerning streaming dynamic  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Problem** & **Algorithm** & **Graph Type** & **Update Mode** & **Fully Dynamic** & **Incremental Dynamic** & **Decmental Dynamic** & **Streaming Dynamic** \\ \hline Vertex coloring & [PERSON] [52] & General & Batch & ✓ & ✓ & ✓ & ✗ \\ \((\Delta+1)\)-coloring & [PERSON] [54] & General & Single & \(O(\log\Delta)\) & ✓ & ✓ & ✗ \\ \((1+\epsilon(1))\Delta\)-coloring & [PERSON] [54] & General & Single & \(O(polylog\Delta)\) & ✓ & ✓ & ✗ \\ \(O(\alpha\log^{2}n)\)-coloring & [PERSON] [49] & General & Single & \(\tilde{O}(polytoga)\) & ✓ & ✓ & ✗ \\ Vertex coloring & [PERSON] [51] & General & Batch & ✓ & ✓ & ✓ & ✗ \\ Vertex coloring & [PERSON] [50] & General & Batch (distributed) & ✓ & ✓ & ✓ & ✗ \\ Vertex coloring & [PERSON] [53] & General & Single & ✓ & ✓ & ✓ & ✗ \\ \hline \hline \end{tabular}
\end{table}
Table 3: Comparison of time complexity for dynamic vertex coloring algorithms.  
algorithms and parallel or distributed algorithms, highlighting an area that demands further investigation and advancement.