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.

We determine the colored patterns that appear in any 2-edge coloring of Kn,n, with n large enough and with sufficient edges in each color. We prove the existence of a positive integer z2 such that any 2-edge coloring of Kn,n with at least z2 edges in each color contains at least one of these patterns. We give a general upper bound for z2 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.