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I am looking for previous work regarding graphs whose vertices can be assigned colours (not necessarily a proper colouring) in such a way that each colour class induces a tree. In particular I am ...

Suppose we have a planar graf with vertices $v_o, \ldots, v_n$, where $n$ is even such that if we checkerboard-color regions in the complement, then the black regions are $n$ (non-degenerated) ...