  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 $\lim_{n \to
\infty}\phi(G_4(n))/n = 1/3$.
