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:
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?