Induced 2-degenerate Subgraphs of Triangle-free Planar Graphs

TL;DR

This paper establishes lower bounds on the size of maximum induced 2-degenerate subgraphs in triangle-free planar graphs, providing formulas based on vertices, edges, and degree constraints.

Contribution

It introduces new bounds for induced 2-degenerate subgraphs in triangle-free planar graphs, extending understanding of their structure and size.

Findings

lpha_2(G)   rac{6n - m - 1}{5} for connected graphs

lpha_2(G)   rac{4}{5}n by Euler's formula

lpha_2(G)   rac{7}{8}n - 18 n_3 for graphs with degree constraints

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 $\alpha_2(G) \geq \frac{7}{8}n - 18 n_3$.