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Let's start with a cellular decomposition of 3-space, a "foam". A foam can be represented by its dual graph: cells and faces are dual to vertices and links.

What about the opposite?

It is clear that one can construct graphs for which no dual foam exists: take a large foam and connect two far-distant, inner vertices with a new link; as this link crosses many cells of the original foam the graph has no dual cellular decomposition in 3-space.

Is there a theorem which says something about higher dimensions?

One can emdedd any graph in 3-space, but for the dual foam this does no longer work. Is there a result showing that the dual foam would live in higher dimensional space? Is anything known about the required number of dimensions for the dual foam? Does it always exist?

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# Graphs, embeddings and dimensions

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