Summary: Let \(k\) be an odd natural number \(\geq 5\), and let \(G\) be a \((6k-7)\)-edge-connected graph of bipartite index at least \(k-1\). Then, for each mapping \(f : V(G) \to \mathbb{N}\), \(G\) has a subgraph \(H\) such that each vertex \(v\) has \(H\)-degree \(f(v)\) modulo \(k\). We apply this to prove that, if \(c\colon V(G)\to \mathbb{Z}_k\) is a proper vertex-coloring of a graph \(G\) of chromatic number \(k \geq 5\) or \(k-1 \geq 6\), then each edge of \(G\) can be assigned a weight 1 or 2 such that each weighted vertex-degree of \(G\) is congruent to \(c\) modulo \(k\). Consequently, each nonbipartite \((6k-7)\)-edge-connected graph of chromatic number at most \(k\) (where \(k\) is any odd natural number \(\geq 3\)) has an edge-weighting with weights 1,2 such that neighboring vertices have distinct weighted degrees (even after reducing these weighted degrees modulo \(k\)). We characterize completely the bipartite graph having an edge-weighting with weights 1,2 such that neighboring vertices have distinct weighted degrees. In particular, that problem belongs to P while it is NP-complete for nonbipartite graphs. The characterization also implies that every 3-edge-connected bipartite graph with at least 3 vertices has such an edge-labeling, and so does every simple bipartite graph of minimum degree at least 3.