Edge colourings and topological graph polynomials

Joanna A. Ellis-Monaghan, Louis H. Kauffman, Iain Moffatt

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A k-valuation is a special type of edge k-colouring of a medial graph. Various graph polynomials, such as the Tutte, Penrose, Bollob\'as-Riordan, and transition polynomials, admit combinatorial interpretations and evaluations as weighted counts of k-valuations. In this paper, we consider a multivariate generating function of k-valuations. We show that this is a polynomial in k and hence defines a graph polynomial. We then show that the resulting polynomial has several desirable properties, including a recursive deletion-contraction-type definition, and specialises to the graph polynomials mentioned above. It also offers an alternative extension of the Penrose polynomial from plane graphs to graphs in other surfaces.
Original languageEnglish
Pages (from-to)290-305
Number of pages16
JournalAustralasian Journal of Combinatorics
Issue number2
Publication statusPublished - Oct 2018


  • math.CO
  • 05C31

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