A Tutte polynomial for non-orientable maps

doi:10.1016/j.endm.2017.07.001

Electronic Notes in Discrete Mathematics

Volume 61, August 2017, Pages 513-519

https://doi.org/10.1016/j.endm.2017.07.001 Get rights and content

Abstract

We construct a new polynomial invariant of maps (graphs embedded in closed surfaces, not necessarily orientable). Our invariant is tailored to contain as evaluations the number of local flows and local tensions taking non-identity values in any given finite group. Moreover, it contains as specializations the Krushkal polynomial, the Bollobás-Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial of the under-lying graph of the map.

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There are more references available in the full text version of this article.

Cited by (5)

Tutte's dichromate for signed graphs
2021, Discrete Applied Mathematics
We introduce the trivariate Tutte polynomial of a signed graph as an invariant of signed graphs up to vertex switching that contains among its evaluations the number of proper colorings and the number of nowhere-zero flows. In this, it parallels the Tutte polynomial of a graph, which contains the chromatic polynomial and flow polynomial as specializations. The number of nowhere-zero tensions (for signed graphs they are not simply related to proper colorings as they are for graphs) is given in terms of evaluations of the trivariate Tutte polynomial at two distinct points. Interestingly, the bivariate dichromatic polynomial of a biased graph, shown by Zaslavsky to share many similar properties with the Tutte polynomial of a graph, does not in general yield the number of nowhere-zero flows of a signed graph. Therefore the “dichromate” for signed graphs (our trivariate Tutte polynomial) differs from the dichromatic polynomial (the rank–size generating function).
The trivariate Tutte polynomial of a signed graph can be extended to an invariant of ordered pairs of matroids on a common ground set — for a signed graph, the cycle matroid of its underlying graph and its frame matroid form the relevant pair of matroids. This invariant is the canonically defined Tutte polynomial of matroid pairs on a common ground set in the sense of a recent paper of Krajewski, Moffatt and Tanasa, and was first studied by Welsh and Kayibi as a four-variable linking polynomial of a matroid pair on a common ground set.
A Tutte polynomial for maps II: The non-orientable case
2020, European Journal of Combinatorics
We construct a new polynomial invariant of maps (graphs embedded in a closed surface, orientable or non-orientable), which contains as specializations the Krushkal polynomial, the Bollobás—Riordan polynomial, the Las Vergnas polynomial, and their extensions to non-orientable surfaces, and hence in particular the Tutte polynomial. Other evaluations include the number of local flows and local tensions taking non-identity values in a given finite group.
A tutte polynomial for maps
2018, Combinatorics Probability and Computing
Tutte's dichromate for signed graphs
2019, arXiv
A Tutte polynomial for maps II: The non-orientable case
2018, arXiv

¹: Supported by Project ERCCZ LL1201 Cores and Czech Science Foundation GA ČR 16-19910S.

²: Supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n° 339109.

³: Supported by a NWO Veni grant.

⁴: Supported by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement n° 339109.

View full text

A Tutte polynomial for non-orientable maps

Abstract

J. Combin. Theory Ser. B

A polynomial of graphs on surfaces

Math. Ann.

A quasi-tree expansion of the Krushkal polynomial

Adv. Appl. Math.

A note on counting flows in signed graphs

Über die reellen Darstellungen der endlichen Gruppen, Sitzungsberichte Königl

Preuss. Akad. Wiss. Berlin

A Tutte polynomial for maps

Character theory of finite groups

Courier Corp.