Increasing the chromatic number of a random graph

What is the minimum number of edges that have to be added to the random graph $G=G_{n,0.5}$ in order to increase its chromatic number $\chi=\chi(G)$ by one percent ? One possibility is to add all missing edges on a set of $1.01 \chi$ vertices, thus creating a clique of chromatic number $1.01 \chi$. This requires, with high probability, the addition of $\Omega(n^2/\log^2 n)$ edges. We show that this is tight up to a constant factor, consider the question for more general random graphs $G_{n,p}$ with $p=p(n)$, and study a local version of the question as well. The question is motivated by the study of the resilience of graph properties, initiated by the second author and Vu, and improves one of their results.