New results on large induced forests in graphs

For a graph $G$, let $a(G)$ denote the maximum size of a subset of vertices that induces a forest. We prove the following. 1. Let $G$ be a graph of order $n$, maximum degree $\Delta>0$ and maximum clique size $\omega$. Then $$a(G) \geq \frac{6n}{2\Delta + \omega +2}.$$ This bound is sharp for cliques. 2. Let $G=(V,E)$ be a triangle-free graph and let $d(v)$ denote the degree of $v \in V$. Then $$a(G) \geq \sum_{v \in V} \min\left(1, \frac{3}{d(v)+2} \right).$$ As a corollary we have that a triangle-free graph $G$ of order $n$, with $m$ edges and average degree $d \geq 2$ satisfies $$a(G) \geq \frac{3n}{d+2}.$$ This improves the lower bound $n - \frac{m}{4}$ of Alon-Mubayi-Thomas for graphs of average degree greater than $4$. Furthermore it improves the lower bound $\frac{20n - 5m - 5}{19}$ of Shi-Xu for (connected) graphs of average degree at least $\frac{9}{2}$.