  In this paper, we consider a weighted generalization of the chromatic number
of a Binomial random graph~\(G.\) We equip each edge with a random weight and
then colour the vertices in such a way that the absolute colour difference
between any two adjacent vertices is at least as large as their edge weight. We
show that with high probability, the weighted colouring number grows linearly
with the maximum vertex degree if the edge weights have sufficiently large
moments. Conversely, if the edge weight moments are unbounded then the weighted
chromatic number is much larger than the maximum vertex degree, with high
probability. We also obtain a sharp threshold result for locally correct
weighted colourings for balanced subgraphs of~\(G.\)
