Title of article :
A Penrose polynomial for embedded graphs
Joanna A. Ellis-Monaghan، نويسنده , , Joanna A. and Moffatt، نويسنده , , Iain، نويسنده ,
Issue Information :
روزنامه با شماره پیاپی سال 2013
We extend the Penrose polynomial, originally defined only for plane graphs, to graphs embedded in arbitrary surfaces. Considering this Penrose polynomial of embedded graphs leads to new identities and relations for the Penrose polynomial which cannot be realized within the class of plane graphs. In particular, by exploiting connections with the transition polynomial and the ribbon group action, we find a deletion–contraction-type relation for the Penrose polynomial. We relate the Penrose polynomial of an orientable chequerboard colourable graph to the circuit partition polynomial of its medial graph and use this to find new combinatorial interpretations of the Penrose polynomial. We also show that the Penrose polynomial of a plane graph G can be expressed as a sum of chromatic polynomials of twisted duals of G . This allows us to obtain a new reformulation of the Four Colour Theorem.
Journal title :
European Journal of Combinatorics