Maximum acyclic and fragmented sets in regular graphs

We show that a typical d-regular graph G of order n does not contain an induced forest with around 2 ln d d n vertices, when n d 1, this bound being best possible because of a result of Frieze and Łuczak [6]. We then deduce an affirmative answer to an open question of Edwards and Farr (see [4]) about fragmentability, which concerns large subgraphs with components of bounded size. An alternative, direct answer to the question is also given. © 2007 Wiley Periodicals, Inc. J Graph Theory 57: 149–156, 2008

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