Degenerate and star colorings of graphs on surfaces

AbstractWe 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 G is a graph of maximum degree Δ, then G admits a degenerate star coloring using O(Δ3/2) colors. We use this result to prove that every graph of genus g admits a degenerate star coloring with O(g3/5) colors. It is also shown that these results are sharp up to a logarithmic factor

3-degenerate induced subgraph of a planar graph

A graph $G$ is $d$-degenerate if every non-null subgraph of $G$ has a vertex of degree at most $d$. We prove that every $n$-vertex planar graph has a $3$-degenerate induced subgraph of order at least $3n/4$.Comment: 28 page