The 3-flow conjecture, factors modulo k, and the 1-2-3-conjecture

Let k be an odd natural number ≥5, and let G   be a (6k−7)(6k−7)-edge-connected graph of bipartite index at least k−1k−1. Then, for each mapping f:V(G)→Nf:V(G)→N, G has a subgraph H such that each vertex v has H  -degree f(v)f(v) modulo k  . We apply this to prove that, if c:V(G)→Zkc:V(G)→Zk is a proper vertex-coloring of a graph G   of chromatic number k≥5k≥5 or k−1≥6k−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)(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 1,21,2 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 1,21,2 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.