An embedding of a metric graph
$(G,d)$
on a closed hyperbolic surface is essential if each complementary region has a negative Euler characteristic. We show, by construction, that given any metric graph, its metric can be rescaled so that it admits an essential and isometric embedding on a closed hyperbolic surface. The essential genus
$g_{e}(G)$
of
$(G,d)$
is the lowest genus of a surface on which such an embedding is possible. We establish a formula to compute
$g_{e}(G)$
and show that, for every integer
$g\geq g_{e}(G)$
, there is an embedding of
$(G,d)$
(possibly after a rescaling of
$d$
) on a surface of genus
$g$
. Next, we study minimal embeddings where each complementary region has Euler characteristic
$-1$
. The maximum essential genus
$g_{e}^{\max }(G)$
of
$(G,d)$
is the largest genus of a surface on which the graph is minimally embedded. We describe a method for an essential embedding of
$(G,d)$
, where
$g_{e}(G)$
and
$g_{e}^{\max }(G)$
are realised.
EMBEDDING OF METRIC GRAPHS ON HYPERBOLIC SURFACES
