Graphs of large girth with prescribed partial circular colourings

This paper completes the constructive proof of the following result: Suppose p/q >= 2 is a rational number, A is a finite set and f(1), f(2,)...,f(n) are mappings from A to {0, 1,...,p - 1}. Then for any integer g, there is a graph G = (V, E) of girth at least g with A subset of V, such that G has exactly n (p, q)-colourings (up to equivalence) g(1), g(2),...,g(n), and each g(i) is an extension of f(i). A probabilistic proof of this result was given in [8]. A constructive proof of the case p/q >= 3 was given in [7].