Maximum 4-degenerate subgraph of a planar graph

Abstract

A graph G is k-degenerate if it can be transformed into an empty graph by subsequent removals of vertices of degree k or less. We prove that every connected planar graph with average degree d≥2 has a 4-degenerate induced subgraph containing at least (38−d)/36 of its vertices. This shows that every planar graph of order n has a 4-degenerate induced subgraph of order more than 8/9⋅n. We also consider a local variation of this problem and show that in every planar graph with at least 7 vertices, deleting a suitable vertex allows us to subsequently remove at least 6 more vertices of degree four or less.