Illinois Journal of Mathematics

Tutte relations, TQFT, and planarity of cubic graphs

Ian Agol and Vyacheslav Krushkal

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It has been known since the work of Tutte that the value of the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ has a number of remarkable properties. We investigate to what extent Tutte’s relations characterize planar graphs. A version of the Tutte linear relation for the flow polynomial at $(3-\sqrt{5})/2$ is shown to give a planarity criterion for $3$-connected cubic (trivalent) graphs. A conjecture is formulated that the golden identity for the flow polynomial characterizes planarity of cubic graphs as well. In addition, Tutte’s upper bound on the chromatic polynomial of planar triangulations at $(3+\sqrt{5})/2$ is generalized to other Beraha numbers, and an exponential lower bound is given for the value at $(3-\sqrt{5})/2$. The proofs of these results rely on the structure of the Temperley–Lieb algebra and more generally on methods of topological quantum field theory.

Article information

Illinois J. Math., Volume 60, Number 1 (2016), 273-288.

Received: 22 December 2015
Revised: 3 November 2016
First available in Project Euclid: 21 June 2017

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 05C31: Graph polynomials 57R56: Topological quantum field theories
Secondary: 57M15: Relations with graph theory [See also 05Cxx] 05C10: Planar graphs; geometric and topological aspects of graph theory [See also 57M15, 57M25]


Agol, Ian; Krushkal, Vyacheslav. Tutte relations, TQFT, and planarity of cubic graphs. Illinois J. Math. 60 (2016), no. 1, 273--288. doi:10.1215/ijm/1498032033.

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