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Evaluations of Topological Tutte Polynomials

Published online by Cambridge University Press:  10 October 2014

J. ELLIS-MONAGHAN  and
I. MOFFATT
J. ELLIS-MONAGHAN
Affiliation:
Department of Mathematics, Saint Michael's College, 1 Winooski Park, Colchester, VT 05439, USA (e-mail: jellis-monaghan@smcvt.edu)
I. MOFFATT
Affiliation:
Department of Mathematics, Royal Holloway, University of London, Egham, Surrey, TW20 0EX, UK (e-mail: iain.moffatt@rhul.ac.uk)
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Abstract

We find new properties of the topological transition polynomial of embedded graphs, Q(G). We use these properties to explain the striking similarities between certain evaluations of Bollobás and Riordan's ribbon graph polynomial, R(G), and the topological Penrose polynomial, P(G). The general framework provided by Q(G) also leads to several other combinatorial interpretations these polynomials. In particular, we express P(G), R(G), and the Tutte polynomial, T(G), as sums of chromatic polynomials of graphs derived from G, show that these polynomials count k-valuations of medial graphs, show that R(G) counts edge 3-colourings, and reformulate the Four Colour Theorem in terms of R(G). We conclude with a reduction formula for the transition polynomial of the tensor product of two embedded graphs, showing that it leads to additional relations among these polynomials and to further combinatorial interpretations of P(G) and R(G).

Keywords

Primary 05C31Secondary 05C10
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 24 , Issue 3 , May 2015 , pp. 556 - 583
Copyright
Copyright © Cambridge University Press 2014 

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