  An embedded cubic graph consisting of segments of geodesics such that the
angles at any vertex are equal to $2\pi/3$ is a closed local minimal net. This
net is regular if all segments of geodesics are equal. The problem of
classification of closed local minimal nets on surfaces of constant negative
curvature has been formulated in the context of the famous Plateau problem in
the one-dimensional case.
  In this paper we prove an asymptotic for $\sharp (W^r(g))$ as $g\to +\infty$
where $g$ is genus and $W^r(g)$ is the set of the regular single-face minimal
nets on surfaces of curvature -1. Then we construct some examples of $f$-face
regular nets, $f>1$.
