A Tutte Polynomial for Coloured Graphs

  • Authors:

  • Béla Bollobás;Oliver Riordan

  • Affiliations:

  • Dept. of Math. Sci., Univ. of Memphis, Memphis TN (bollobas@msci.memphis.edu) and Dept. of Pure Math. and Math. Stats., Univ. of Cambridge, England (B.Bollobas@dpmms.cam.ac.uk) and Inst. for Adv. ...;Dept. of Math. Sci., Univ. of Memphis, TN (bollobas@msci.memphis.edu) and Dept. of Pure Math. and Math. Stat., Univ. of Cambridge, England (B.Bollobas@dpmms.cam.ac.uk)

  • Venue:
  • Combinatorics, Probability and Computing
  • Year:
  • 1999

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Abstract

We define a polynomial W on graphs with colours on the edges, by generalizing the spanning tree expansion of the Tutte polynomial as far as possible: we give necessary and sufficient conditions on the edge weights for this expansion not to depend on the order used. We give a contraction-deletion formula for W analogous to that for the Tutte polynomial, and show that any coloured graph invariant satisfying such a formula can be obtained from W. In particular, we show that generalizations of the Tutte polynomial obtained from its rank generating function formulation, or from a random cluster model, can be obtained from W. Finally, we find the most general conditions under which W gives rise to a link invariant, and give as examples the one-variable Jones polynomial, and an invariant taking values in ℤ/22ℤ.