Colouring Non-sparse Random Intersection Graphs

An intersection graph of n vertices assumes that each vertex is equipped with a subset of a global label set. Two vertices share an edge when their label sets intersect. Random Intersection Graphs (RIGs) (as defined in [18, 31]) consider label sets formed by the following experiment: each vertex, independently and uniformly, examines all the labels (m in total) one by one. Each examination is independent and the vertex succeeds to put the label in her set with probability p. Such graphs nicely capture interactions in networks due to sharing of resources among nodes. We study here the problem of efficiently coloring (and of finding upper bounds to the chromatic number) of RIGs. We concentrate in a range of parameters not examined in the literature, namely: (a) m = n for α less than 1 (in this range, RIGs differ substantially from the ErdösRenyi random graphs) and (b) the selection probability p is quite high (e.g. at least ln 2 n m in our algorithm) and disallows direct greedy colouring methods. We manage to get the following results: – For the case mp ≤ β lnn, for any constant β < 1− α, we prove that np colours are enough to colour most of the vertices of the graph with high probability (whp). This means that even for quite dense graphs, using the same number of colours as those needed to properly colour the clique induced by any label suffices to colour almost all of the vertices of the graph. Note also that this range of values of m, p is quite wider than the one studied in [4]. – We propose and analyze an algorithm CliqueColour for finding a proper colouring of a random instance of Gn,m,p, for any mp ≥ ln n. The algorithm uses information of the label sets assigned to the vertices of Gn,m,p and runs in O (