Aggregating the preferences of self-interested agents is a key problem for multiagent systems, and one general method for doing so is to vote over the alternatives (candidates). Unfortunately, the Gibbard-Satterthwaite theorem shows that when there are three or more candidates, all reasonable voting rules are manipulable (in the sense that there exist situations in which a voter would benefit f...

A proper coloring of a graph G = (V,E) is called optimal if the number of colors used is the minimal possible; i.e., it coincides with the chromatic number of G. We investigate the typical behavior of the number of distinct optimal colorings of a random graph G(n, p), for various values of the edge probability p = p(n). Our main result shows that for every constant 1/3 < a < 2, most of the grap...

We undertook a retrospective study of all isolated mandibular fractures which had required active management over a 1-year period at the Maxillofacial Unit at Newcastle General Hospital. Patients with single or multiple fractures of the mandible were included in the study, if there were other simultaneous fractures of the facial skeleton, those patients were excluded. All case notes and radiogr...