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This question regards colourings on edges and vertices on countable directed multigraphs. We start with an example. Let $G=\mathbb Z^2$. We define two functions $a_h$ and $a_v$ from $\mathbb Z^2$ to $\...

Assign to the $n$ nodes of a path graph vertex weights forming a permutation of $(0,\ldots,n{-}1)$. Now iterate the following update repeatedly: Each node sums the weights of its neighbors, and that ...