Centre for Discrete and Applicable Mathematics

 CDAM Research Report, LSE-CDAM-2000-07

June 2000

The chromatic roots of generalised dodecahedra

Norman Biggs and Philipp Reinfeld


The aim of this paper is to study the chromatic polynomials of a family of cubic graphs  Dn  with  4n  vertices. In particular  D5  is the graph of the regular dodecahedron.
We use the compatibility matrix method to obtain a third-order recursion for the leading terms of the chromatic polynomial of  Dn,  as well as information about the remaining terms. Then we apply a theorem of Beraha, Kahane and Weiss to describe the curves on which the limit points of the chromatic roots lie.
In Section 6 we analyse a method of Salas and Sokal, in order to provide a good description of the limit curves.

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