The Sierpinski gasket is not a good example for this because of the bound you saw on the degree of the graph. I'd venture to say that this is because the gasket falls into a class of fractals called post-critically finite. It is the fact that in pcf fractals when level n cells intersect there are a uniformly bounded number of them. You might be interested in looking at finitely ramified but not post-critically finite fractals. See this for some nice pictures of the Diamond fractal. Since the number of level n-cells intersecting is unbounded in n you can embed graphs without a bound on degree.
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The Sierpinski gasket is not a good example for this because of the bound you saw on the degree of the graph. I'd venture to say that this is because the gasket falls into a class of fractals called post-critically finite. It is the fact that in pcf fractals when level n cells intersect there are a uniformly bounded number of them. You might be interested in looking at finitely ramified but not post-critically finite fractals. See this for some nice pictures of the Diamond fractal. Since the number of level n-cells intersecting is unbounded in n you can embed graphs without a bound on degree. |
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