On Tutte's chromatic invariant
Authors:
Sabin Cautis and David M. Jackson
Journal:
Trans. Amer. Math. Soc. 362 (2010), 491507
MSC (2000):
Primary 05C15
Published electronically:
August 18, 2009
MathSciNet review:
2550161
Fulltext PDF Free Access
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Abstract: Consider a simple connected graph embedded in the plane together with a contractible circuit . For a partition of the vertex set of we denote by the number of ways of assigning one of given colours to each vertex of so that vertices in the same block of have the same colour. Tutte showed that this polynomial may be expressed uniquely as a linear combination of over all planar partitions of , with scalars that are independent of . We show that the (chromatic) invariants have a natural algebraic setting in terms of the orthogonal projection from the partition algebra to the TemperleyLieb subalgebra . We define the genus of a partition and give an extension of the invariants to arbitrary genus . Finally, we summarise the rôle of the genus 0 invariants in the algebraic approach of Birkhoff and Lewis to the Four Colour Theorem.
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Additional Information
Sabin Cautis
Affiliation:
Department of Mathematics, Rice University, Houston, Texas 77251
Email:
scautis@math.harvard.edu
David M. Jackson
Affiliation:
Department of Combinatorics and Optimization, University of Waterloo, Ontario, Canada N2L 3G1
Email:
dmjackson@math.uwaterloo.ca
DOI:
http://dx.doi.org/10.1090/S0002994709048363
Keywords:
Chromatic invariant,
noncrossing partitions,
TemperleyLieb algebra,
partition algebra,
BirkhoffLewis equations
Received by editor(s):
February 1, 2006
Received by editor(s) in revised form:
July 11, 2007, and May 9, 2008
Published electronically:
August 18, 2009
Article copyright:
© Copyright 2009
American Mathematical Society
