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.
Highlights
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
A graph G is k-degenerate if every subgraph of G has a vertex of degree k or less
Borodin [4] proved that every planar graph G is acyclically 5-colourable, that is, we can partition the vertices of G into five classes such that the subgraph induced by the union of any two classes is acyclic
Summary
A graph G is k-degenerate if every subgraph of G has a vertex of degree k or less. Equivalently, a graph is k-degenerate if we can delete the whole graph by subsequently removing vertices of degree at most k. Degeneracy 3 was studied by Oum and Zhu [7] who were interested in the order of a maximum 4-choosable induced subgraph of a planar graph. The best we are aware of is that both octahedron and icosahedron contain no induced 3-degenerate subgraph of order greater than 5/6 · |V (G)|. The problem of maximum degenerate subgraphs was studied for general graphs by Alon, Kahn, and Seymour [2] They precisely determined how large k-degenerate induced subgraph one can guarantee depending only on the degree sequence of G. Every connected planar graph with average degree d 2 has a 4-degenerate induced subgraph containing at least (38 − d)/36 of its vertices. The collected vertices induce a 4-degenerate subgraph of G containg more than 8/9 of its vertices These results are probably not the best possible. In every planar graph with at least 12 vertices we can delete a vertex in such a way that we can collect at least 11 vertices
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
