The vertex-colouring {a,b}-edge-weighting problem is NP-complete for every pair of weights
Let G be a graph. From an edge-weighting w : E(G) -> {a,b} of G such that a and b are two distinct real numbers, one obtains a vertex-colouring chi_w of G defined as chi_w(u) = sum_{v in N(u)} w(uv) for every u in V(G). If chi_w is a proper colouring of G, i.e. two adjacent vertices of G receive distinct colours by chi_w, then we say that w is vertex-colouring. We investigate the complexity of the problem of deciding whether a graph admits a vertex-colouring edge-weighting taking values among a given pair {a,b}, which is already known to be NP-complete when {a,b} is either {0,1} or {1,2}. We show this problem to be NP-complete for every pair of real weights.