  We describe a new random greedy algorithm for generating regular graphs of
high girth: Let $k\geq 3$ and $c \in (0,1)$ be fixed. Let $n \in \mathbb{N}$ be
even and set $g = c \log_{k-1} (n)$. Begin with a Hamilton cycle $G$ on $n$
vertices. As long as the smallest degree $\delta (G)<k$, choose, uniformly at
random, two vertices $u,v \in V(G)$ of degree $\delta(G)$ whose distance is at
least $g-1$. If there are no such vertex pairs, abort. Otherwise, add the edge
$uv$ to $E(G)$.
  We show that with high probability this algorithm yields a $k$-regular graph
with girth at least $g$. Our analysis also implies that there are $\left(
\Omega (n) \right)^{kn/2}$ labeled $k$-regular $n$-vertex graphs with girth at
least $g$.
