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

Planar graphs have two-coloring number at most 8

We prove that the two-coloring number of any planar graph is at most 8. This resolves a question of Kierstead et al. (2009). The result is optimal