For a graph G and not necessarily proper k-edge coloring c:E(G)→{1,…,k}, let mi(G) be the number of edges color i, call color-balanced if mi(G)=mj(G) for every two colors i j. Several famous open problems relate to this notion; Ryser's conjecture on transversals in latin squares, instance, is equivalent statement that properly n-edge colored complete bipartite Kn,n has perfect matching. We cont...