= Face colouring
A face colouring of a <plane graph> assigns colours to its <faces of a plane graph> so that distinct faces sharing an <edge> have different colours. Equivalently it is a <graph colouring> of the <dual graph>, disregarding dual loops that arise when an original bridge is incident to only one face on both sides. Isolated contact at a <vertex> does not impose a colour constraint.
For the <dual graph> of a plane <tree>, the face-adjacency graph is the original <tree>, because taking the geometric dual twice recovers the original embedding on the sphere. Every nontrivial <tree> is a <bipartite graph>, so its dual requires exactly two face colours, even though that dual has only one vertex and all its <edges> are loops. A <graph drawing> must retain the nesting of those loops to preserve the correct face-adjacency graph.
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