Summary: 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 = Gn,0.5
in order to increase its chromatic number = (G) by one percent ? One possibility is to add all
missing edges on a set of 1.01 vertices, thus creating a clique of chromatic number 1.01. This
requires, with high probability, the addition of (n2
n) edges. We show that this is tight up
to a constant factor, consider the question for more general random graphs Gn,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.
Consider the probability space whose points are graphs on n labeled vertices, where each pair of
vertices forms an edge, randomly and independently with probability p. The random graph Gn,p
denotes a random point in this probability space. This concept is one of the central notions in modern
discrete mathematics and it has been studied intensively during the last 50 years. By now, there are
thousands of papers and two excellent monographs by Bollob´as  and by Janson et al.  devoted