Colouring Non-sparse Random Intersection Graphs

Sotiris E. Nikoletseas, Christoforos L. Raptopoulos, Paul G. Spirakis

Abstract: 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 [18,32]) 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 α for α less than 1 (in this range, RIGs differ substantially from the Erdös-Renyi random graphs) and (b) the selection probability p is quite high (e.g. at least \(\frac{\ln^2{n}}{m}\) in our algorithm) and disallows direct greedy colouring methods.