  Given a graph $G$, a 2-coloring of the edges of $K_n$ is said to contain a
balanced copy of $G$ if we can find a copy of $G$ such that half of its edges
are in each color class. If, for every sufficiently large $n$, there exists an
integer $k$ such that every 2-coloring of $K_n$ with more than $k$ edges in
each color class contains a balanced copy of $G$, then we say that $G$ is
balanceable. Balanceability was introduced by Caro, Hansberg and Montejano, who
also gave a structural characterization of balanceable graphs.
  In this paper, we extend the study of balanceability by finding new
sufficient conditions for a graph to be balanceable or not. We use those
conditions to fully characterize the balanceability of graph classes such as
rectangular and triangular grids, as well as a special class of circulant
graphs.
