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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