  In the first part of this paper, we consider weighted domination in the case
where the vertices of the complete graph on~\(n\) vertices are equipped with
independent and identically distributed (i.i.d.) weights. We use the
probabilistic iteration to determine sufficient conditions for maximizing the
weighted domination probability. In the second part, we study a weighted
generalization of the chromatic number and estimate the minimum number of
colours needed to satisfy the constraints when the weights themselves are
random. We show that the "extra" cost incurred for weighted colouring is small
if the weights have sufficiently large moments. We also consider inhomogenous
random graphs where the edge probabilities are not necessarily all same and
obtain bounds for the chromatic number in terms of its averaged edge
probabilities.
