Maximum 4-Degenerate Subgraph of a Planar Graph

Robert Lukot'ka, Ján Mazák, Xuding Zhu


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.


Degenerate subgraphs; Planar graphs; List colouring

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