2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162484We 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.: If $G$ is a graph of maximum degree $Δ$, then $G$ admits a degenerate star coloring using $O(Δ^{3/2})$ colors. We use this result to prove that every graph of genus $g$ admits a degenerate star coloring with $O(g^{3/5})$ colors. It is also shown that these results are sharp up to a logarithmic factor.16 pages, 2 figuresCombinatorics05C15text