Performance ratio of the generalized greedy algorithm for $q$-coloring problem

In this paper a generalized greedy algorithm finding $q$-colorable subgraph is described, where $q$ is a natural number, and it is proven that the performance ratio of the described algorithm does not exceed $\dfrac{e}{e-1}$ for arbitrary $q$. Then it is shown that the performance ratio of the simple greedy algorithm also doesn't exceed $\dfrac{e}{e-1}$. Thus, it follows that the performance ratio $2$ previously found for simple greedy algorithm is not attainable.