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A Tutte Polynomial for Maps

Published online by Cambridge University Press:  12 April 2018

ANDREW GOODALL ,
THOMAS KRAJEWSKI ,
GUUS REGTS  and
LLUÍS VENA
ANDREW GOODALL
Affiliation:
Computer Science Institute, Charles University, Prague, Czech Republic (e-mail: andrew@iuuk.mff.cuni.cz)
THOMAS KRAJEWSKI
Affiliation:
Aix-Marseille Université, Université de Toulon, CNRS, CPT, Marseille, France (e-mail: krajew@cpt.univ-mrs.fr)
GUUS REGTS
Affiliation:
Department of Mathematics, University of Amsterdam, Amsterdam, Netherlands (e-mail: guusregts@gmail.com)
LLUÍS VENA
Affiliation:
Computer Science Institute, Charles University, Prague, Czech Republic (e-mail: andrew@iuuk.mff.cuni.cz) Faculty of Science, University of Amsterdam, Amsterdam, Netherlands (e-mail: lluis.vena@gmail.com)
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Abstract

We follow the example of Tutte in his construction of the dichromate of a graph (i.e. the Tutte polynomial) as a unification of the chromatic polynomial and the flow polynomial in order to construct a new polynomial invariant of maps (graphs embedded in orientable surfaces). We call this the surface Tutte polynomial. The surface Tutte polynomial of a map contains the Las Vergnas polynomial, the Bollobás–Riordan polynomial and the Krushkal polynomial as specializations. By construction, the surface Tutte polynomial includes among its evaluations the number of local tensions and local flows taking values in any given finite group. Other evaluations include the number of quasi-forests.

Keywords

Primary 05C31Secondary 05C1057M15

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 27 , Issue 6 , November 2018 , pp. 913 - 945
Copyright
Copyright © Cambridge University Press 2018 

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Footnotes

†

Supported by Project ERCCZ LL1201 Cores and the Czech Science Foundation, GA ČR 16-19910S.

‡

Supported by ANR grant ANR JCJC ‘CombPhysMat2Tens’.

§

Supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 339109, and by an NWO Veni grant.

Supported by the Center of Excellence–Institute for Theoretical Computer Science, Prague, P202/12/G061, and by Project ERCCZ LL1201 Cores.

‖

The research leading to these results received funding from the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement 339109.

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