Let k be an odd natural number ≥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)→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:V(G)→Zk is a proper vertex-coloring of a graph G of chromatic number k≥5 or k−1≥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 ≥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.
- Factors modulo k