Communications in Mathematical Physics

, Volume 354, Issue 1, pp 31–84 | Cite as

Counting Coloured Planar Maps: Differential Equations

  • Olivier Bernardi
  • Mireille Bousquet-Mélou


We address the enumeration of q-coloured planar maps counted by the number of edges and the number of monochromatic edges. We prove that the associated generating function is differentially algebraic, that is, satisfies a non-trivial polynomial differential equation with respect to the edge variable. We give explicitly a differential system that characterizes this series. We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings. In statistical physics terms, we solve the q-state Potts model on random planar lattices. This work follows a first paper by the same authors, where the generating function was proved to be algebraic for certain values of q, including \({q=1, 2}\) and 3. It is known to be transcendental in general. In contrast, our differential system holds for an indeterminate q. For certain special cases of combinatorial interest (four colours; proper q-colourings; maps equipped with a spanning forest), we derive from this system, in the case of triangulations, an explicit differential equation of order 2 defining the generating function. For general planar maps, we also obtain a differential equation of order 3 for the four-colour case and for the self-dual Potts model.


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© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsBrandeis UniversityWalthamUSA
  2. 2.CNRS, LaBRI, Université de BordeauxTalenceFrance

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