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Are you asking about lower or upper bounds on the integrality gap? For lower bounds, we know that the gap is at least 4/3. That example that shows this is planar, graphic, and Euclidean (well each of these is a special case of the other). So the lower bound is the same for all of them. In terms of upper bounds, as @Gamwow said, there is Gutekunst and ...

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