Average Distance in Coloured Graphs Peter Dankelmann 1 School of Mathematical and Statistical Sciences University of Natal, Durban, 4041, South Africa Wayne Goddard 1 School of Geological and Computer Sciences University of Natal, Durban, 4041, South Africa Peter Slater 1 Department of Mathematics and Statistics University of Alabama in Huntsville, Huntsville AL 35899, USA Abstract For a graph G where the vertices are coloured, the coloured distance of G is defined as the sum of the distances between all unordered pairs of vertices having different colours. Then for a fixed supply s of colours, d s (G) is defined as the minimum coloured distance over all colourings with s. This generalizes the concepts of median and average distance. In this paper we explore bounds on this parameter especially a natural lower bound and the particular case of balanced 2-colourings (equal numbers of red and blue). We show that the general problem is NP-hard but there is a polynomial-time algorithm for trees. 1 Introduction Hulme and Slater [6] introduced the following facilities location problem. Given a connected graph G =(V (G),E(G)) on n vertices, some number k of facilities is specified. The facilities are to be placed on the graph. At each vertex there is to be either a person or a facility. Each person must visit each facility (perhaps because each facility is different). The question is how to place these facilities. That is, one must place the facilities to minimize the sum over all pairs (u, v), where u is a facility and v is not, of the distance d(u, v) (measured as the number of edges in the shortest path connecting them). The placement of the facilities may be thought of as a colouring of the vertices with two colours. This generalizes to “coloured distance ”. A colouring of a graph 1 Research supported in part by South African Foundation for Research Development 1