A Better Bound on the Largest Induced Forests in Triangle-Free Planar Graph

Abstract

It is well-known that there exists a triangle-free planar graph of n vertices that has the largest induced forest of order at most $$\frac{5n}{8}$$ . Salavatipour (Graphs Comb 22(1):113–126, 2006) proved that there is a forest of order at least $$\frac{5n}{9.41}$$ in any triangle-free planar graph of n vertices. Dross et al. (Large induced forests in planar graphs with girth 4 or 5, arXiv:1409.1348 , 2014) improved Salavatipour’s bound to $$\frac{5n}{9.17}$$ . In this work, we further improve the bound to $$\frac{5n}{9}$$ . Our technique is inspired by the recent ideas from Lukot’ka et al. (Electron J Comb 22(1):P1–P11, 2015).