Decycling numbers of random regular graphs

The decycling number phi(G) of a graph G is the smallest number of vertices which can be removed from C so that the resultant graph contains no cycles. In this paper, we study the decycling numbers of random regular graphs. For a random cubic graph G of order n, we prove that phi(G) = [n/4 + 1/2] holds asymptotically almost surely. This is the result of executing a greedy algorithm for decycling G making use of a randomly chosen Hamilton cycle. For a general random d-regular graph G of order n, where d greater than or equal to 4, we prove that phi(G)/n can be bounded below and above asymptotically almost surely by certain constants b(d) and B(d), depending solely on d, which are determined by solving, respectively, an algebraic equation and a system of differential equations. (C) 2002 Wiley Periodicals, Inc.