  We determine the colored patterns that appear in any $2$-edge coloring of
$K_{n,n}$, with $n$ large enough and with sufficient edges in each color. We
prove the existence of a positive integer $z_2$ such that any $2$-edge coloring
of $K_{n,n}$ with at least $z_2$ edges in each color contains at least one of
these patterns. We give a general upper bound for $z_2$ and prove its tightness
for some cases. We define the concepts of bipartite $r$-tonality and bipartite
omnitonality using the complete bipartite graph as a base graph. We provide a
characterization for bipartite $r$-tonal graphs and prove that every tree is
bipartite omnitonal. Finally, we define the bipartite balancing number and
provide the exact bipartite balancing number for paths and stars.
