Induced 2-degenerate Subgraphs of Triangle-free Planar Graphs

09/12/2017

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by   Zdeněk Dvořák, et al.

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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 α_2(G). We prove that if G is a connected triangle-free planar graph with n vertices and m edges, then α_2(G) ≥6n - m - 1/5. By Euler's Formula, this implies α_2(G) ≥4/5n. 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 α_2(G) ≥7/8n - 18 n_3.