On the performance guarantee of First Fit for sum coloring

In sum coloring, it is required to find a proper coloring of the vertices of a graph using positive integers, such that the sum of colors of the vertices is minimized. First Fit is the natural coloring algorithm that processes vertices one by one (in some order), and colors the current vertex with the minimum color that was not used for any of its already considered neighbors. Here, we study the approximation ratio of First Fit for sum coloring for the class of claw-free graphs, its most natural subclasses, and its generalizations, and get tight bounds for it.