Matched approximation bound for the sum of a greedy coloring

In the minimum sum coloring problem, the goal is to color the vertices of a graph with the positive integers such that the sum of all colors is minimal. Iteratively coloring maximum independent sets has been shown to yield a 4+o(1) approximation for the minimum sum coloring problem. In this note, we show that this bound is tight, by constructing a graph for which the approximation ratio of this coloring is 4-o(1).