Decycling numbers of random regular graphs

Authors:
Sheng Bau;Nicholas C. Wormald;Sanming Zhou
Affiliations:
School of Mathematics, Statistics and Information Technology, University of Natal, Pietermaritzburg, South Africa;Department of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia;Department of Mathematics and Statistics, University of Melbourne, VIC 3010, Australia
Venue:
Random Structures & Algorithms - Special issue: Proceedings of the tenth international conference "Random structures and algorithms"
Year:
2002

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Abstract

The decycling number φ(G) of a graph G is the smallest number of vertices which can be removed from G 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 φ(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 ≥ 4, we prove that φ(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.