Comparing First-Fit and Next-Fit for online edge coloring

We study the performance of the algorithms First -Fit and Next -Fit for two online edge coloring problems. In the min-coloring problem, all edges must be colored using as few colors as possible. In the max-coloring problem, a fixed number of colors is given, and as many edges as possible should be colored. Previous analysis using the competitive ratio has not separated the performance of First -Fit and Next -Fit, but intuition suggests that First -Fit should be better than Next -Fit. We compare First -Fit and Next -Fit using the relative worst-order ratio, and show that First -Fit is better than Next -Fit for the min-coloring problem. For the max-coloring problem, we show that First -Fit and Next -Fit are not strictly comparable, i.e., there are graphs for which First -Fit is significantly better than Next -Fit and graphs where Next -Fit is slightly better than First -Fit.