 
Summary: Large induced forests in sparse graphs
Noga Alon
, Dhruv Mubayi
, Robin Thomas
February 22, 2002
Abstract
For a graph G, let a(G) denote the maximum size of a subset of vertices that induces
a forest. Suppose that G is connected with n vertices, e edges, and maximum degree
. Our results include:
(a) if 3, and G = K4, then a(G) ne/41/4 and this is sharp for all permissible
e 3 (mod 4),
(b) if 3, then a(G) (G) + (n  (G))/(  1)2.
Several problems remain open.
1 Introduction
For a (simple, undirected) graph G = (V, E), we say that an S V is an acyclic set if the
induced subgraph G[S] is a forest. We let a(G) denote the maximum size of an acyclic set
in G. In [4], the minimum possible value of a(G) is determined, where G ranges over all
Department of Mathematics, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel
Aviv University, Tel Aviv, Israel, email: noga@math.tau.ac.il. Research supported in part by a
