  Lov\'asz conjectured that every connected 4-regular planar graph G admits a
realization as a system of circles, i.e., it can be drawn on the plane
utilizing a set of circles, such that the vertices of G correspond to the
intersection and touching points of the circles and the edges of G are the arc
segments among pairs of intersection and touching points of the circles. In
this paper, we settle this conjecture. In particular, (a) we first provide
tight upper and lower bounds on the number of circles needed in a realization
of any simple 4-regular planar graph, (b) we affirmatively answer Lov\'asz's
conjecture, if G is 3-connected, and (c) we demonstrate an infinite class of
simple connected 4-regular planar graphs which are not 3-connected (i.e.,
either simply connected or biconnected) and do not admit realizations as a
system of circles.
