Induced 2-Degenerate Subgraphs of Triangle-Free Planar Graphs

Abstract

A graph is $k$-degenerate if every subgraph has minimum degree at most $k$.  We provide lower bounds on the size of a maximum induced 2-degenerate subgraph in a triangle-free planar graph.  We denote the size of a maximum induced 2-degenerate subgraph of a graph $G$ by $\alpha_2(G)$.  We prove that if $G$ is a connected triangle-free planar graph with $n$ vertices and $m$ edges, then $\alpha_2(G) \geq \frac{6n - m - 1}{5}$.  By Euler's Formula, this implies $\alpha_2(G) \geq \frac{4}{5}n$.  We also prove that if $G$ is a triangle-free planar graph on $n$ vertices with at most $n_3$ vertices of degree at most three, then $\alpha_2(G) \geq \frac{7}{8}n - 18 n_3$.