We consider the Chromatic Sum Problem on bipartite graphs which appears to be much harder than the classical Chromatic Number Problem. We prove that the Crmatic Sum Problem is NP-complete on planar bipartite graphs with ∇≤ 5, but polynomial on bipartite graphs with ∇≤ 3, for which we construct an O(n ²)-time algorithm. Hence, we tighten the borderline of intractability for this problem on bipartite graphs with bounded degree, namely: the case ∇ = 3 is easy, ∇ = 5 is hard. Moreover, we construct a 27/26-approximation algorithm for this problem thus improving the best known approximation ratio of 10/9.