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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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##### References:
 [1] Alon, J Graph Theory 38 pp 113– (2001) [2] Bafna, SIAM J Discrete Math 12 pp 289– (1999) [3] Bau, Austr J Combinat 25 pp 285– (2002) [4] Bau, Utilitas Math 59 pp 129– (2001) [5] Beineke, J Graph Theory 25 pp 59– (1997) [6] Random graphs, Academic Press, London, 1985. [7] Bondy, Can Math Bull 30 pp 193– (1987) [8] Britikov, Mat Zametki 43 pp 672– (1988) [9] Graph theory, Springer-Verlag, New York, 1997. [10] Duckworth, Random Struct Alg [11] Erd?s, J Combinat Theory 41B pp 61– (1986) [12] ?Reducibility among combinatorial problems,? Complexity of computer computations, Eds. and 85-103, Plenum Press, New York, London, 1972. [13] Kim, J Combinat Theory Ser B 81 pp 20– (2001) [14] Kirchhoff, Ann Phys Chem 72 pp 497– (1847) [15] Li, Acta Math Sci (Engl Ed) 19 pp 375– (1999) [16] Liang, Inf Process Lett 52 pp 123– (1994) [17] Liang, Acta Inf 34 pp 337– (1997) [18] Liu, Discrete Math 148 pp 119– (1996) [19] ?Concentration,? Probabilistic methods for algorithmic discrete mathematics, Eds. et al., 195-248, Springer-Verlag, Berlin, 1998. [20] ?Asymptotic enumeration methods,? Handbook of Combinatorics, Vol. II, Eds. and Elsevier, Amsterdam, 1995. [21] Robinson, Random Struct Alg 3 pp 117– (1992) [22] Robinson, Random Struct Alg 5 pp 363– (1994) [23] Ruci?ski, J Austr Math Soc 72 pp 67– (2002) [24] Ueno, Discrete Math 72 pp 355– (1988) [25] Wormald, Ann Appl Probab 5 pp 1217– (1995) [26] ?The differential equation method for random graph processes and greedy algorithms,? Lectures on approximation and randomized algorithms, Eds. and 73-155. PWN, Warsaw, 1999. [27] ?Models of random regular graphs,? Surveys in combinatorics, 1999, Eds. and London Mathematical Society Lecture Notes Series, Vol. 267, 239-298. Cambridge University Press, Cambridge, 1999. [28] Analysis of greedy algorithms on graphs with bounded degrees, Discrete Math, to appear. · Zbl 1029.05147 [29] Zheng, Discrete Math 85 pp 89– (1990)
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