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. (2004) [8]: If GG is a graph of maximum degree ΔΔ, then GG admits a degenerate star coloring using O(Δ3/2)O(Δ3/2) colors. We use this result to prove that every graph of genus gg admits a degenerate star coloring with O(g3/5)O(g3/5) colors. It is also shown that these results are sharp up to a logarithmic factor.