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 ${\alpha }_2(G)\ge \frac{6n-m-1}{5}$. By Euler's Formula, this implies ${\alpha }_2(G)\ge \frac{4}{5}n$. 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 ${\alpha }_2(G)\ge \frac{7}{8}n-18n_3$.

17 pages, 3 figures -- v2 revised according to referee comments