Maximum 4degenerate 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 $d \ge 2$ has a 4degenerate induced subgraph containing at least $(38d)/36$ of its vertices. This shows that every planar graph of order $n$ has a 4degenerate 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.
 Publication:

arXiv eprints
 Pub Date:
 May 2013
 DOI:
 10.48550/arXiv.1305.6195
 arXiv:
 arXiv:1305.6195
 Bibcode:
 2013arXiv1305.6195L
 Keywords:

 Mathematics  Combinatorics;
 Computer Science  Discrete Mathematics