Maximum 4-degenerate subgraph of a planar graph

TL;DR

This paper proves that every connected planar graph contains a large 4-degenerate induced subgraph, with size proportional to the total number of vertices, and explores local vertex removal strategies to find such subgraphs.

Contribution

It establishes a lower bound on the size of 4-degenerate induced subgraphs in planar graphs and introduces a local vertex removal method for constructing them.

Findings

Every connected planar graph with average degree d ≥ 2 has a 4-degenerate induced subgraph with at least (38-d)/36 of its vertices.

Any planar graph with at least 7 vertices allows removal of a vertex to enable the removal of at least 6 vertices of degree ≤4.

A planar graph of order n has a 4-degenerate induced subgraph of size more than 8/9 of n.

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 $d \ge 2$ 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 $8/9 \cdot n$. 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.