A dessin d'enfant is a graph, with its vertices colored alternately black and white, embedded in ... a plane. For the coloring to exist, the graph must be bipartite. ... The ... embedding may be described combinatorially using a rotation system...

Any dessin can provide the surface it is embedded in with a structure as a Riemann surface. It is natural to ask which Riemann surfaces arise in this way.

The wikipage on Dessin d'enfants, also gives an example:

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The dessin d'enfant arising from the rational function $f = −(x − 1)^3(x − 9)/64x$. Not to scale.

Further a table, listing the degrees of vertices, is given. Later Shabat polynomials are mentioned, which are the corresponding ones for trees.

My question: Is it possible to assign a rational function to any kind of bipartite planar cubic graph?

Or more specific: How to construct the polynomial for the chamfered cube?

  • $\begingroup$ I have no idea about any of this, but I just want to say that I am very surprised no other users have answered or even commented on this post. I apologise that I cannot help you, and also on their behalf. $\endgroup$ – Mr Pie Mar 21 '18 at 6:50
  • $\begingroup$ @user477343 thanks, while waitin' I found a good portion of answers myself. Stay tuned... $\endgroup$ – draks ... Mar 21 '18 at 9:01
  • $\begingroup$ Hahah :P ${}{}$ $\endgroup$ – Mr Pie Mar 21 '18 at 9:04

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