It is fairly well know when graphs can be embedded on various surfaces. Also, it is not hard to see that any graph can be embedded in 3-dimensional space. Has anyone ever studied the embeddability of graphs on various fractals? If so, are there any interesting results?

For example, I made some quick deductions about graphs on Sierpinski's Gasket. No graph that has vertices of degree greater than 4 can be embedded on Sierpinski's Gasket. Also, I conjecture that one can not embed $K_4$ into Sierpinski's Gasket either.

Also, one might ask questions like, "Is it true that one can embed any graph on any space with Hausdorff dimension that is greater than or equal to 3?"

These are just some curiosities that came to me and a friend of mine earlier today, and I am curious if the Math Overflow community knows anything about such things.