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Decycling numbers of random regular graphs. (English) Zbl 1012.05099
Summary: The decycling number \(\phi(G)\) 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 \(\phi (G) = \lceil n/4 + 1/2\rceil\) 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 \(d\geq 4\), 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.

MSC:
05C38 Paths and cycles
05C80 Random graphs (graph-theoretic aspects)
05C35 Extremal problems in graph theory
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