  One of the keys to the success of multimessenger astronomy is the rapid
identification of the electromagnetic wave counterpart, kilonova (KN), of the
gravitational-wave (GW) event. Despite its importance, it is hard to find a KN
associated with a GW event, due to a poorly constrained GW localization map and
numerous signals that could be confused as a KN. Here, we present the
Gravitational-wave Electromagnetic wave Counterpart Korean Observatory (GECKO)
project, the GECKO observation of GW190425, and prospects of GECKO in the
fourth observing run (O4) of the GW detectors. We outline our follow-up
observation strategies during O3. In particular, we describe our
galaxy-targeted observation criteria that prioritize based on galaxy
properties. Armed with this strategy, we performed an optical and/or
near-infrared follow-up observation of GW190425, the first binary neutron star
merger event during the O3 run. Despite a vast localization area of 7460 deg^2,
we observed 621 host galaxy candidates, corresponding to 29.5% of the scores we
assigned, with most of them observed within the first 3 days of the GW event.
Ten transients were discovered during this search, including a new transient
with a host galaxy. No plausible KN was found, but we were still able to
constrain the properties of potential KNe using upper limits. The GECKO
observation demonstrates that GECKO can possibly uncover a GW170817-like KN at
a distance less than 200 Mpc if the localization area is of the order of
hundreds of square degrees, providing a bright prospect for the identification
of GW electromagnetic wave counterparts during the O4 run.
