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Suppose a generalization of planar graph is considered into 3D space :

a graph is said "spatial" if it can be constructed in Euclidean 3D space in such a way that no edge intersects a face.

The questions are the following :

-as for plane graphs their chromatic number is 4, can we show that the chromatic number for spatial graphs is 5 since K5 is spatial but not K6. (Kn is the complete n-vertices graph)

-If we consider a further generalization, can it be shown that Kn is "constructible" in a n-2 dimensional space such that no edge intersects any n-1-hyperface, but not K(n+1) ?

a graph is said "spatial" if it can be constructed in Euclidean 3D space in such a way that no edge intersects a face.

The questions are the following :

-as for plane graphs their chromatic number is 4, can we show that the chromatic number for spatial graphs is 5 since K5 is spatial but not K6. (Kn is the complete n-vertices graph)

-If we consider a further generalization, can it be shown that Kn is "constructible" in a n-2 dimensional space such that no edge intersects any n-1-hyperface, but not K(n+1) ?

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