Tight upper bound on the number of optimal paths in weighted coloured-edge graphs

A weighted coloured-edge is a graph for which each edge is assigned both a positive weight and a discrete colour, and can be used to model transportation and computer networks in which there are multiple transportation modes. In such a graph paths are compared by their weight in each colour, resulting in a Pareto set of optimal paths from one vertex to another. This paper will give a tight upper bound on the cardinality of the Pareto set and explain some results toward establishing the average case cardinality