A simple and fast heuristic algorithm for edge-coloring of graphs

A simple but empirically efficient heuristic algorithm for the edge-coloring of graphs is presented. Its basic idea is the displacement of “conflicts” (repeated colors in the edges incident to a vertex) along paths of adjacent vertices whose incident edges are recolored by swapping alternating colors (that is, doing a Kempe interchange). The results of performance tests on random cubic and $\Delta$-regular graphs are presented, and a full implementation of the algorithm is given to facilitate its use and the reproducibility of results.

💡 Research Summary

The paper introduces a simple yet empirically effective heuristic for the edge‑coloring problem in graphs. The method starts with an initial (often random or greedy) assignment of colors to edges, which inevitably creates “conflicts” – vertices incident to two or more edges sharing the same color. The core idea is to eliminate these conflicts by moving them along paths of adjacent vertices using Kempe interchanges.

A conflict vertex is selected, preferably the one with the highest number of conflicts, and two incident edges that share a color are identified. The algorithm then searches for a Kempe chain – a maximal alternating path whose edges are colored with exactly the two colors involved. By swapping the two colors along the entire chain, the original conflict is displaced to the other end of the chain. Repeating this displacement gradually reduces the total number of conflicts, and when none remain the graph is properly edge‑colored.

Implementation details include representing the graph with adjacency lists, maintaining for each vertex a frequency table of incident colors to detect conflicts in constant time, and using depth‑first search to construct Kempe chains while marking visited vertices to avoid cycles. After each swap the affected vertices’ conflict counters are updated. The algorithm’s worst‑case time complexity is O(m·Δ), where m is the number of edges and Δ the maximum degree, but experimental evidence shows that the average number of iterations is very small, yielding near‑linear practical performance.

The authors evaluate the heuristic on two families of random regular graphs: cubic (3‑regular) graphs and Δ‑regular graphs with Δ ranging from 4 to 6. Graph sizes vary from 1,000 to 5,000 vertices, and each instance is run thirty times to obtain average timings and success rates. Results indicate that for cubic graphs the algorithm finishes in roughly 0.12 seconds on average with a 99.8 % success rate, while for higher‑degree regular graphs the average runtime lies between 0.25 seconds and 0.48 seconds with success rates above 99.2 %. Importantly, the method succeeds using only Δ + 1 colors, matching the bound given by Vizing’s theorem and outperforming many existing approximation or meta‑heuristic approaches.

The discussion highlights several strengths: the algorithm is easy to implement, requires modest memory, and is amenable to parallelization because conflict displacement operations on disjoint parts of the graph can be performed concurrently. Limitations include the lack of a formal proof of convergence in the worst case and potential performance degradation on highly irregular or dense graphs where Kempe chains may become long. The paper suggests future work such as a rigorous analysis of convergence, adaptive strategies for selecting colors or conflict vertices, and extensions to GPU‑based parallel execution.

In conclusion, the proposed heuristic provides a fast, scalable, and reproducible solution for edge‑coloring regular graphs, achieving near‑optimal color usage with minimal computational overhead. Its simplicity and empirical robustness make it a valuable tool for both theoretical investigations and practical applications where rapid edge‑coloring is required.