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The concepts of (k, d)-coloring and the star chromatic number, studied by Vince, by Bondy and Hell, and by Zhu are shown to reflect the cographic instance of a wider concept, that of fractional nowhere-zero flows in regular matroids.

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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) → Z k 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-labelling, and so does every simple bipartite graph of minimum degree at least 3.

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