Degenerate and star colorings of graphs on surfaces

We study the degenerate, the star and the degenerate star chromatic numbers and their relation to the genus of graphs. As a tool we prove the following strengthening of a result of Fertin et al. [8]: If G is a graph of maximum degree ∆, then G admits a degenerate star coloring using O(∆) colors. We use this result to prove that every graph of genus g admits a degenerate star coloring with O(g) colors. It is also shown that these results are sharp up to a logarithmic factor.