Chromatic polynomials and toroidal graphs

The chromatic polynomials of some families of quadrangulations of the torus can be found explicitly. The method, known as ‘bracelet theory’ is based on a decomposition in terms of representations of the symmetric group. The results are particularly appropriate for studying the limit curves of the chromatic roots of these families. In this paper these techniques are applied to a family of quadrangulations with chromatic number 3, and a simple parametric equation for the limit curve is obtained. The results are in complete agreement with experimental evidence.