Counting coloured planar maps: differential equations

Understand

We address the enumeration of q-coloured planar maps counted bythe 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 withrespect to the edge variable.
• We give explicitly a differential systemthat characterizes this series.
• We then prove a similar result for planar triangulations, thus generalizing a result of Tutte dealing with their proper q-colourings.