On the Decycling Number of $4$-regular Random Graphs

Lyuben Lichev; Dieter Mitsche

On the Decycling Number of $4$-regular Random Graphs

Probability 2020-05-20 v2

Authors: Lyuben Lichev , Dieter Mitsche

Abstract

The decycling number $\phi(G)$ of a graph $G$ is the smallest number of vertices which can be removed from $G$ so that the resulting graph has no cycles. Bau, Wormald and Zhou conjectured that with probability tending to one the decycling number of the random $4$ -regular graph $G_4(n)$ on $n$ vertices is equal to $\lceil (n+1)/3\rceil$ . In this paper we show that this conjecture holds asymptotically, i.e. asymptotically almost surely $\lim_{n \to \infty}\phi(G_4(n))/n = 1/3$ .

Keywords

graph cycles graph pebbling random graphs

Cite

@article{arxiv.2003.00538,
  title  = {On the Decycling Number of $4$-regular Random Graphs},
  author = {Lyuben Lichev and Dieter Mitsche},
  journal= {arXiv preprint arXiv:2003.00538},
  year   = {2020}
}

Comments

There is an error in Observation 3.3 in the paper, leading to false conclusions later on

On the Decycling Number of $4$-regular Random Graphs

Abstract

Keywords

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