The forced colouring function of a graph

The forced colouring function of a graph gives the probability that a random assignment of colours to a random subset of vertices can be extended, by a simple local process called forcing, to give a proper colouring of the whole graph using the same set of available colours. This is a polynomial for each fixed number of colours, and was introduced as a subject for research on the general theory of graph polynomials. In this paper we establish its fundamental properties and give combinatorial interpretations of its derivatives at two particular points. We also prove that the problem of computing its value at any specific point in a certain interval is #P-hard.