First-fit chromatic numbers of -degenerate graphs

The first-fit chromatic number of a graph is the number of colors needed in the worst case of a greedy coloring. In this short note, we first give counterexamples to some results by Balogh et al. (2008) [1], and then prove that every n -vertex d -degenerate graph has first-fit chromatic number at most log d + 1 d n + 2 .